Table of Contents Part I Ordinary Differential Equations 1 Introduction to Differential Equations 1 2 First-Order Differential Equations 22 3 Higher-Order Differential Equations 99 4 The Laplace Transform 198 5 Series Solutions of Linear Differential Equations 252 6 Numerical Solutions of Ordinary Differential Equations 317 Part II Vectors, Matrices, and Vector Calculus 7 Vectors 339 8 Matrices 373 9 Vector Calculus 438 Part III Systems of Differential Equations 10 Systems of Linear Differential Equations 551 11 Systems of Nonlinear Differential Equations 604 Part IV Fourier Series and Partial Differential Equations 12 Orthogonal Functions and Fourier Series 634 13 Boundary-Value Problems in Rectangular Coordinates 680 14 Boundary-Value Problems in Other Coordinate Systems 755 15 Integral Transform Method 793 16 Numerical Solutions of Partial Differential Equations 832 Part V Complex Analysis 17 Functions of a Complex Variable 854 18 Integration in the Complex Plane 877 19 Series and Residues 896 20 Conformal Mappings 919 Appendices Appendix II Gamma function 942 Projects 3.7 Road Mirages 944 3.10 The Ballistic Pendulum 946 8.1 Two-Ports in Electrical Circuits 947 8.2 Traffic Flow 948 8.15 Temperature Dependence of Resistivity 949 9.16 Minimal Surfaces 950 14.3 The Hydrogen Atom 952 15.4 The Uncertainity Inequality in Signal Processing 955 15.4 Fraunhofer Diffraction by a Circular Aperture 958 16.2 Instabilities of Numerical Methods 960 Part I Ordinary Differential Equations Introduction to Differential Equations 1 EXERCISES 1.1 Definitions and Terminology 1. Second order; linear 2. Third order; nonlinear because of (dy/dx)4 3. Fourth order; linear 4. Second order; nonlinear because of cos(r + u) 5. Second order; nonlinear because of (dy/dx)2 or 6. Second order; nonlinear because of R2 ! 1 + (dy/dx)2 7. Third order; linear 8. Second order; nonlinear because of ẋ2 9. Writing the differential equation in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y because of y 2 . However, writing it in the form (y 2 − 1)(dx/dy) + x = 0, we see that it is linear in x. 10. Writing the differential equation in the form u(dv/du) + (1 + u)v = ueu we see that it is linear in v. However, writing it in the form (v + uv − ueu )(du/dv) + u = 0, we see that it is nonlinear in u. 11. From y = e−x/2 we obtain y ′ = − 21 e−x/2 . Then 2y ′ + y = −e−x/2 + e−x/2 = 0. 12. From y = 6 5 − 65 e−20t we obtain dy/dt = 24e−20t , so that dy + 20y = 24e−20t + 20 dt " 6 6 −20t − e 5 5 # = 24. 13. From y = e3x cos 2x we obtain y ′ = 3e3x cos 2x − 2e3x sin 2x and y ′′ = 5e3x cos 2x − 12e3x sin 2x, so that y ′′ − 6y ′ + 13y = 0. 14. From y = − cos x ln(sec x + tan x) we obtain y ′ = −1 + sin x ln(sec x + tan x) and y ′′ = tan x + cos x ln(sec x + tan x). Then y ′′ + y = tan x. 15. The domain of the function, found by solving x + 2 ≥ 0, is [−2, ∞). From y ′ = 1 + 2(x + 2)−1/2 we have (y − x)y ′ = (y − x)[1 + (2(x + 2)−1/2 ] = y − x + 2(y − x)(x + 2)−1/2 = y − x + 2[x + 4(x + 2)1/2 − x](x + 2)−1/2 = y − x + 8(x + 2)1/2 (x + 2)−1/2 = y − x + 8. 1 1.1 Definitions and Terminology An interval of definition for the solution of the differential equation is (−2, ∞) because y ′ is not defined at x = −2. 16. Since tan x is not defined for x = π/2 + nπ, n an integer, the domain of y $ $ {x $ 5x ̸= π/2 + nπ} or {x $ x = ̸ π/10 + nπ/5}. From y ′ = 25 sec2 5x we have = 5 tan 5x is y ′ = 25(1 + tan2 5x) = 25 + 25 tan2 5x = 25 + y 2 . An interval of definition for the solution of the differential equation is (−π/10, π/10). Another interval is (π/10, 3π/10), and so on. $ $ 17. The domain of the function is {x $ 4 − x2 ̸= 0} or {x $ x ̸= −2 or x ̸= 2}. From y ′ = 2x/(4 − x2 )2 we have ′ y = 2x " 1 4 − x2 #2 = 2xy. An interval of definition for the solution of the differential equation is (−2, 2). Other intervals are (−∞, −2) and (2, ∞). √ 18. The function is y = 1/ 1 − sin x , whose domain is obtained from 1 − sin x ̸= 0 or sin x ̸= 1. Thus, the domain $ is {x $ x ̸= π/2 + 2nπ}. From y ′ = − 12 (1 − sin x)−3/2 (− cos x) we have 2y ′ = (1 − sin x)−3/2 cos x = [(1 − sin x)−1/2 ]3 cos x = y 3 cos x. An interval of definition for the solution of the differential equation is (π/2, 5π/2). Another one is (5π/2, 9π/2), and so on. 19. Writing ln(2X −1)−ln(X −1) = t and differentiating implicitly we obtain X 2 1 dX dX − =1 2X − 1 dt X − 1 dt " # 2 dX 1 − =1 2X − 1 X − 1 dt 4 2 2X − 2 − 2X + 1 dX =1 (2X − 1)(X − 1) dt -4 dX = −(2X − 1)(X − 1) = (X − 1)(1 − 2X). dt Exponentiating both sides of the implicit solution we obtain -2 2 4 t -2 -4 2X − 1 = et X −1 2X − 1 = Xet − et (et − 1) = (et − 2)X X= et − 1 . et − 2 Solving et − 2 = 0 we get t = ln 2. Thus, the solution is defined on (−∞, ln 2) or on (ln 2, ∞). The graph of the solution defined on (−∞, ln 2) is dashed, and the graph of the solution defined on (ln 2, ∞) is solid. 2 1.1 Definitions and Terminology 20. Implicitly differentiating the solution, we obtain y dy dy −2x2 − 4xy + 2y =0 dx dx −x2 dy − 2xy dx + y dy = 0 4 2 2xy dx + (x2 − y)dy = 0. Using the quadratic formula to solve y 2 − 2x2 y − 1 = 0 for y, we get √ √ % & y = 2x2 ± 4x4 + 4 /2 = x2 ± x4 + 1 . Thus, two explicit solutions √ √ are y1 = x2 + x4 + 1 and y2 = x2 − x4 + 1 . Both solutions are defined on (−∞, ∞). The graph of y1 (x) is solid and the graph of y2 is dashed. -4 -2 2 4 x -2 -4 21. Differentiating P = c1 et / (1 + c1 et ) we obtain dP c1 et [(1 + c1 et ) − c1 et ] (1 + c1 et ) c1 et − c1 et · c1 et = = 2 dt 1 + c1 et 1 + c1 et (1 + c1 et ) = 2 22. Differentiating y = e−x ) x c1 et 1 + c1 et ' 1− 2 ( c1 et = P (1 − P ). 1 + c1 et 2 et dt + c1 e−x we obtain 0 ′ −x2 x2 y =e e −x2 − 2xe ) x t2 −x2 e dt − 2c1 xe 0 −x2 = 1 − 2xe 0 Substituting into the differential equation, we have ) x ) ′ −x2 t2 −x2 −x2 y + 2xy = 1 − 2xe e dt − 2c1 xe + 2xe 0 23. From y = c1 e2x + c2 xe2x we obtain x ) x 2 2 et dt − 2c1 xe−x . 2 2 et dt + 2c1 xe−x = 1. 0 d2 y dy = (2c1 + c2 )e2x + 2c2 xe2x and = (4c1 + 4c2 )e2x + 4c2 xe2x , so that dx dx2 d2 y dy −4 + 4y = (4c1 + 4c2 − 8c1 − 4c2 + 4c1 )e2x + (4c2 − 8c2 + 4c2 )xe2x = 0. 2 dx dx 24. From y = c1 x−1 + c2 x + c3 x ln x + 4x2 we obtain dy = −c1 x−2 + c2 + c3 + c3 ln x + 8x, dx d2 y = 2c1 x−3 + c3 x−1 + 8, dx2 and d3 y = −6c1 x−4 − c3 x−2 , dx3 so that x3 d3 y d2 y dy + 2x2 2 − x + y = (−6c1 + 4c1 + c1 + c1 )x−1 + (−c3 + 2c3 − c2 − c3 + c2 )x 3 dx dx dx + (−c3 + c3 )x ln x + (16 − 8 + 4)x2 = 12x2 . 25. From y = * −x2 , 2 x , x<0 x≥0 we obtain y ′ = * −2x, 2x, x<0 so that xy ′ − 2y = 0. x≥0 3 1.1 Definitions and Terminology 26. The function y(x) is not continuous at x = 0 since lim− y(x) = 5 and lim+ y(x) = −5. Thus, y ′ (x) does not x→0 x→0 exist at x = 0. 27. (a) From y = emx we obtain y ′ = memx . Then y ′ + 2y = 0 implies memx + 2emx = (m + 2)emx = 0. Since emx > 0 for all x, m = −2. Thus y = e−2x is a solution. (b) From y = emx we obtain y ′ = memx and y ′′ = m2 emx . Then y ′′ − 5y ′ + 6y = 0 implies m2 emx − 5memx + 6emx = (m − 2)(m − 3)emx = 0. Since emx > 0 for all x, m = 2 and m = 3. Thus y = e2x and y = e3x are solutions. 28. (a) From y = xm we obtain y ′ = mxm−1 and y ′′ = m(m − 1)xm−2 . Then xy ′′ + 2y ′ = 0 implies xm(m − 1)xm−2 + 2mxm−1 = [m(m − 1) + 2m]xm−1 = (m2 + m)xm−1 = m(m + 1)xm−1 = 0. Since xm−1 > 0 for x > 0, m = 0 and m = −1. Thus y = 1 and y = x−1 are solutions. (b) From y = xm we obtain y ′ = mxm−1 and y ′′ = m(m − 1)xm−2 . Then x2 y ′′ − 7xy ′ + 15y = 0 implies x2 m(m − 1)xm−2 − 7xmxm−1 + 15xm = [m(m − 1) − 7m + 15]xm = (m2 − 8m + 15)xm = (m − 3)(m − 5)xm = 0. Since xm > 0 for x > 0, m = 3 and m = 5. Thus y = x3 and y = x5 are solutions. In Problems 29–32, we substitute y = c into the differential equations and use y ′ = 0 and y ′′ = 0 29. Solving 5c = 10 we see that y = 2 is a constant solution. 30. Solving c2 + 2c − 3 = (c + 3)(c − 1) = 0 we see that y = −3 and y = 1 are constant solutions. 31. Since 1/(c − 1) = 0 has no solutions, the differential equation has no constant solutions. 32. Solving 6c = 10 we see that y = 5/3 is a constant solution. 33. From x = e−2t + 3e6t and y = −e−2t + 5e6t we obtain dx = −2e−2t + 18e6t dt and dy = 2e−2t + 30e6t . dt Then and x + 3y = (e−2t + 3e6t ) + 3(−e−2t + 5e6t ) = −2e−2t + 18e6t = dx dt 5x + 3y = 5(e−2t + 3e6t ) + 3(−e−2t + 5e6t ) = 2e−2t + 30e6t = dy . dt 34. From x = cos 2t + sin 2t + 51 et and y = − cos 2t − sin 2t − 51 et we obtain and 1 dx = −2 sin 2t + 2 cos 2t + et dt 5 and dy 1 = 2 sin 2t − 2 cos 2t − et dt 5 1 d2 x = −4 cos 2t − 4 sin 2t + et 2 dt 5 and d2 y 1 = 4 cos 2t + 4 sin 2t − et . 2 dt 5 Then and 1 1 d2 x 4y + et = 4(− cos 2t − sin 2t − et ) + et = −4 cos 2t − 4 sin 2t + et = 2 5 5 dt 4 1.1 Definitions and Terminology 1 1 d2 y 4x − et = 4(cos 2t + sin 2t + et ) − et = 4 cos 2t + 4 sin 2t − et = 2 . 5 5 dt 35. (y ′ )2 + 1 = 0 has no real solutions because (y ′ )2 + 1 is positive for all functions y = φ(x). 36. The only solution of (y ′ )2 + y 2 = 0 is y = 0, since if y ̸= 0, y 2 > 0 and (y ′ )2 + y 2 ≥ y 2 > 0. 37. The first derivative of f (x) = ex is ex . The first derivative of f (x) = ekx is kekx . The differential equations are y ′ = y and y ′ = ky, respectively. 38. Any function of the form y = cex or y = ce−x is its own second derivative. The corresponding differential equation is y ′′ − y = 0. Functions of the form y = c sin x or y = c cos x have second derivatives that are the negatives of themselves. The differential equation is y ′′ + y = 0. ! ! √ 39. We first note that 1 − y 2 = 1 − sin2 x = cos2 x = | cos x|. This prompts us to consider values of x for which cos x < 0, such as x = π. In this case $ $ $ $ dy $$ d $ (sin x)$ = cos x$x=π = cos π = −1, = $ $ dx $ dx x=π x=π but ! ! √ 1 − y 2 |x=π = 1 − sin2 π = 1 = 1. ! Thus, y = sin x will only be a solution of y ′ = 1 − y 2 when cos x > 0. An interval of definition is then (−π/2, π/2). Other intervals are (3π/2, 5π/2), (7π/2, 9π/2), and so on. 40. Since the first and second derivatives of sin t and cos t involve sin t and cos t, it is plausible that a linear combination of these functions, A sin t + B cos t, could be a solution of the differential equation. Using y ′ = A cos t − B sin t and y ′′ = −A sin t − B cos t and substituting into the differential equation we get y ′′ + 2y ′ + 4y = −A sin t − B cos t + 2A cos t − 2B sin t + 4A sin t + 4B cos t = (3A − 2B) sin t + (2A + 3B) cos t = 5 sin t. Thus 3A − 2B = 5 and 2A + 3B = 0. Solving these simultaneous equations we find A = particular solution is y = 15 13 sin t − 10 13 15 13 cos t. 10 and B = − 13 . A 41. One solution is given by the upper portion of the graph with domain approximately (0, 2.6). The other solution is given by the lower portion of the graph, also with domain approximately (0, 2.6). 42. One solution, with domain approximately (−∞, 1.6) is the portion of the graph in the second quadrant together with the lower part of the graph in the first quadrant. A second solution, with domain approximately (0, 1.6) is the upper part of the graph in the first quadrant. The third solution, with domain (0, ∞), is the part of the graph in the fourth quadrant. 43. Differentiating (x3 + y 3 )/xy = 3c we obtain xy(3x2 + 3y 2 y ′ ) − (x3 + y 3 )(xy ′ + y) =0 x2 y 2 3x3 y + 3xy 3 y ′ − x4 y ′ − x3 y − xy 3 y ′ − y 4 = 0 (3xy 3 − x4 − xy 3 )y ′ = −3x3 y + x3 y + y 4 y′ = y(y 3 − 2x3 ) y 4 − 2x3 y = . 2xy 3 − x4 x(2y 3 − x3 ) 44. A tangent line will be vertical where y ′ is undefined, or in this case, where x(2y 3 − x3 ) = 0. This gives x = 0 and 2y 3 = x3 . Substituting y 3 = x3 /2 into x3 + y 3 = 3xy we get 5 1.1 Definitions and Terminology # " 1 1 x x3 + x3 = 3x 2 21/3 3 3 3 x = 1/3 x2 2 2 x3 = 22/3 x2 x2 (x − 22/3 ) = 0. Thus, there are vertical tangent lines at x = 0 and x = 22/3 , or at (0, 0) and (22/3 , 21/3 ). Since 22/3 ≈ 1.59, the estimates of the domains in Problem 42 were close. √ √ 45. The derivatives of the functions are φ′1 (x) = −x/ 25 − x2 and φ′2 (x) = x/ 25 − x2 , neither of which is defined at x = ±5. 46. To determine if a solution curve passes through (0, 3) we let t = 0 and P = 3 in the equation P = c1 et /(1+c1 et ). This gives 3 = c1 /(1 + c1 ) or c1 = − 32 . Thus, the solution curve P = −3et (−3/2)et = 1 − (3/2)et 2 − 3et passes through the point (0, 3). Similarly, letting t = 0 and P = 1 in the equation for the one-parameter family of solutions gives 1 = c1 /(1 + c1 ) or c1 = 1 + c1 . Since this equation has no solution, no solution curve passes through (0, 1). 47. For the first-order differential equation integrate f (x). For the second-order differential equation integrate twice. + + In the latter case we get y = ( f (x)dx)dx + c1 x + c2 . 48. Solving for y ′ using the quadratic formula we obtain the two differential equations ! ! 1, 1, y′ = and y ′ = 2 + 2 1 + 3x6 2 − 2 1 + 3x6 , x x so the differential equation cannot be put in the form dy/dx = f (x, y). 49. The differential equation yy ′ − xy = 0 has normal form dy/dx = x. These are not equivalent because y = 0 is a solution of the first differential equation but not a solution of the second. 50. Differentiating we get y ′ = c1 + 3c2 x2 and y ′′ = 6c2 x. Then c2 = y ′′ /6x " # " ′′ # xy ′′ y ′ y= y − x+ x3 = xy ′ − 2 6x and c1 = y ′ − xy ′′ /2, so 1 2 ′′ x y 3 and the differential equation is x2 y ′′ − 3xy ′ + 3y = 0. 2 51. (a) Since e−x is positive for all values of x, dy/dx > 0 for all x, and a solution, y(x), of the differential equation must be increasing on any interval. 2 2 dy dy (b) lim = lim e−x = 0 and lim = lim e−x = 0. Since dy/dx approaches 0 as x approaches −∞ x→−∞ x→∞ dx x→∞ x→−∞ dx and ∞, the solution curve has horizontal asymptotes to the left and to the right. (c) To test concavity we consider the second derivative " # 2 d2 y d , −x2 d dy = −2xe−x . = e = dx2 dx dx dx Since the second derivative is positive for x < 0 and negative for x > 0, the solution curve is concave up on (−∞, 0) and concave down on (0, ∞). 6 1.1 (d) Definitions and Terminology y x 52. (a) The derivative of a constant solution y = c is 0, so solving 5 − c = 0 we see that c = 5 and so y = 5 is a constant solution. (b) A solution is increasing where dy/dx = 5−y > 0 or y < 5. A solution is decreasing where dy/dx = 5−y < 0 or y > 5. 53. (a) The derivative of a constant solution is 0, so solving y(a − by) = 0 we see that y = 0 and y = a/b are constant solutions. (b) A solution is increasing where dy/dx = y(a − by) = by(a/b − y) > 0 or 0 < y < a/b. A solution is decreasing where dy/dx = by(a/b − y) < 0 or y < 0 or y > a/b. (c) Using implicit differentiation we compute d2 y = y(−by ′ ) + y ′ (a − by) = y ′ (a − 2by). dx2 Solving d2 y/dx2 = 0 we obtain y = a/2b. Since d2 y/dx2 > 0 for 0 < y < a/2b and d2 y/dx2 < 0 for a/2b < y < a/b, the graph of y = φ(x) has a point of inflection at y = a/2b. (d) y y=aêb y=0 x 54. (a) If y = c is a constant solution then y ′ = 0, but c2 + 4 is never 0 for any real value of c. (b) Since y ′ = y 2 + 4 > 0 for all x where a solution y = φ(x) is defined, any solution must be increasing on any interval on which it is defined. Thus it cannot have any relative extrema. (c) Using implicit differentiation we compute d2 y/dx2 = 2yy ′ = 2y(y 2 + 4). Setting d2 y/dx2 = 0 we see that y = 0 corresponds to the only possible point of inflection. Since d2 y/dx2 < 0 for y < 0 and d2 y/dx2 > 0 for y > 0, there is a point of inflection where y = 0. 7 1.1 Definitions and Terminology y (d) x 55. In Mathematica use Clear[y] y[x ]:= x Exp[5x] Cos[2x] y[x] y''''[x] − 20y'''[x] + 158y''[x] − 580y'[x] +841y[x]//Simplify The output will show y(x) = e5x x cos 2x, which verifies that the correct function was entered, and 0, which verifies that this function is a solution of the differential equation. 56. In Mathematica use Clear[y] y[x ]:= 20Cos[5Log[x]]/x − 3Sin[5Log[x]]/x y[x] xˆ3 y'''[x] + 2xˆ2 y''[x] + 20x y'[x] − 78y[x]//Simplify The output will show y(x) = 20 cos(5 ln x)/x − 3 sin(5 ln x)/x, which verifies that the correct function was entered, and 0, which verifies that this function is a solution of the differential equation. EXERCISES 1.2 Initial-Value Problems 1. Solving −1/3 = 1/(1 + c1 ) we get c1 = −4. The solution is y = 1/(1 − 4e−x ). 2. Solving 2 = 1/(1 + c1 e) we get c1 = −(1/2)e−1 . The solution is y = 2/(2 − e−(x+1) ) . 3. Letting x = 2 and solving 1/3 = 1/(4 + c) we get c = −1. The solution is y = 1/(x2 − 1). This solution is defined on the interval (1, ∞). 4. Letting x = −2 and solving 1/2 = 1/(4 + c) we get c = −2. The solution is y = 1/(x2 − 2). This solution is √ defined on the interval (−∞, − 2 ). 5. Letting x = 0 and solving 1 = 1/c we get c = 1. The solution is y = 1/(x2 + 1). This solution is defined on the interval (−∞, ∞). 8 1.2 Initial-Value Problems 6. Letting x = 1/2 and solving −4 = 1/(1/4 + c) we get c = −1/2. The solution is y = 1/(x2 − 1/2) = 2/(2x2 − 1). √ √ This solution is defined on the interval (−1/ 2 , 1/ 2 ). In Problems 7–10, we use x = c1 cos t + c2 sin t and x′ = −c1 sin t + c2 cos t to obtain a system of two equations in the two unknowns c1 and c2 . 7. From the initial conditions we obtain the system c1 = −1 c2 = 8. The solution of the initial-value problem is x = − cos t + 8 sin t. 8. From the initial conditions we obtain the system c2 = 0 −c1 = 1. The solution of the initial-value problem is x = − cos t. 9. From the initial conditions we obtain Solving, we find c1 = √ √ 3 1 1 c1 + c2 = 2 2 2 √ 1 3 − c1 + c2 = 0. 2 2 3/4 and c2 = 1/4. The solution of the initial-value problem is √ x = ( 3/4) cos t + (1/4) sin t. 10. From the initial conditions we obtain √ √ √ 2 2 c1 + c2 = 2 2 2 √ √ √ 2 2 − c1 + c2 = 2 2 . 2 2 Solving, we find c1 = −1 and c2 = 3. The solution of the initial-value problem is x = − cos t + 3 sin t. In Problems 11–14, we use y = c1 ex + c2 e−x and y ′ = c1 ex − c2 e−x to obtain a system of two equations in the two unknowns c1 and c2 . 11. From the initial conditions we obtain c1 + c2 = 1 Solving, we find c1 = 3 2 c1 − c2 = 2. and c2 = − 12 . The solution of the initial-value problem is y = 23 ex − 21 e−x . 12. From the initial conditions we obtain ec1 + e−1 c2 = 0 Solving, we find c1 = 1 2 ec1 − e−1 c2 = e. and c2 = − 12 e2 . The solution of the initial-value problem is y= 1 x 1 2 −x 1 1 e − e e = ex − e2−x . 2 2 2 2 13. From the initial conditions we obtain e−1 c1 + ec2 = 5 e−1 c1 − ec2 = −5. 9 1.2 Initial-Value Problems Solving, we find c1 = 0 and c2 = 5e−1 . The solution of the initial-value problem is y = 5e−1 e−x = 5e−1−x . 14. From the initial conditions we obtain c1 + c2 = 0 c1 − c2 = 0. Solving, we find c1 = c2 = 0. The solution of the initial-value problem is y = 0. 15. Two solutions are y = 0 and y = x3 . 16. Two solutions are y = 0 and y = x2 . (Also, any constant multiple of x2 is a solution.) ∂f 2 = y −1/3 . Thus, the differential equation will have a unique solution in any ∂y 3 rectangular region of the plane where y ̸= 0. ! √ 18. For f (x, y) = xy we have ∂f /∂y = 12 x/y . Thus, the differential equation will have a unique solution in any 17. For f (x, y) = y 2/3 we have region where x > 0 and y > 0 or where x < 0 and y < 0. 19. For f (x, y) = y ∂f 1 we have = . Thus, the differential equation will have a unique solution in any region x ∂y x where x ̸= 0. 20. For f (x, y) = x + y we have ∂f = 1. Thus, the differential equation will have a unique solution in the entire ∂y plane. 21. For f (x, y) = x2 /(4 − y 2 ) we have ∂f /∂y = 2x2 y/(4 − y 2 )2 . Thus the differential equation will have a unique solution in any region where y < −2, −2 < y < 2, or y > 2. ∂f −3x2 y 2 x2 we have = 2 . Thus, the differential equation will have a unique solution in 1 + y3 ∂y (1 + y 3 ) any region where y ̸= −1. 22. For f (x, y) = ∂f 2x2 y y2 we have = 2 . Thus, the differential equation will have a unique solution in x2 + y 2 ∂y (x2 + y 2 ) any region not containing (0, 0). 23. For f (x, y) = 24. For f (x, y) = (y + x)/(y − x) we have ∂f /∂y = −2x/(y − x)2 . Thus the differential equation will have a unique solution in any region where y < x or where y > x. ! ! y 2 − 9 and ∂f /∂y = y/ y 2 − 9. We see that f and ∂f /∂y are both continuous in the regions of the plane determined by y < −3 and y > 3 with no restrictions on In Problems 25–28, we identify f (x, y) = x. 25. Since 4 > 3, (1, 4) is in the region defined by y > 3 and the differential equation has a unique solution through (1, 4). 26. Since (5, 3) is not in either of the regions defined by y < −3 or y > 3, there is no guarantee of a unique solution through (5, 3). 27. Since (2, −3) is not in either of the regions defined by y < −3 or y > 3, there is no guarantee of a unique solution through (2, −3). 28. Since (−1, 1) is not in either of the regions defined by y < −3 or y > 3, there is no guarantee of a unique solution through (−1, 1). 29. (a) A one-parameter family of solutions is y = cx. Since y ′ = c, xy ′ = xc = y and y(0) = c · 0 = 0. 10 1.2 Initial-Value Problems (b) Writing the equation in the form y ′ = y/x, we see that R cannot contain any point on the y-axis. Thus, any rectangular region disjoint from the y-axis and containing (x0 , y0 ) will determine an interval around x0 and a unique solution through (x0 , y0 ). Since x0 = 0 in part (a), we are not guaranteed a unique solution through (0, 0). (c) The piecewise-defined function which satisfies y(0) = 0 is not a solution since it is not differentiable at x = 0. d 30. (a) Since tan(x + c) = sec2 (x + c) = 1 + tan2 (x + c), we see that y = tan(x + c) satisfies the differential dx equation. (b) Solving y(0) = tan c = 0 we obtain c = 0 and y = tan x. Since tan x is discontinuous at x = ±π/2, the solution is not defined on (−2, 2) because it contains ±π/2. (c) The largest interval on which the solution can exist is (−π/2, π/2). 1 d , 1 1 31. (a) Since − = = y 2 , we see that y = − is a solution of the differential equation. dx x+c (x + c)2 x+c (b) Solving y(0) = −1/c = 1 we obtain c = −1 and y = 1/(1 − x). Solving y(0) = −1/c = −1 we obtain c = 1 and y = −1/(1 + x). Being sure to include x = 0, we see that the interval of existence of y = 1/(1 − x) is (−∞, 1), while the interval of existence of y = −1/(1 + x) is (−1, ∞). 32. (a) Solving y(0) = −1/c = y0 we obtain c = −1/y0 and y=− y0 1 = , −1/y0 + x 1 − y0 x y0 ̸= 0. Since we must have −1/y0 + x ̸= 0, the largest interval of existence (which must contain 0) is either (−∞, 1/y0 ) when y0 > 0 or (1/y0 , ∞) when y0 < 0. (b) By inspection we see that y = 0 is a solution on (−∞, ∞). 33. (a) Differentiating 3x2 − y 2 = c we get 6x − 2yy ′ = 0 or yy ′ = 3x. (b) Solving 3x2 − y 2 = 3 for y we get ! y = φ1 (x) = 3(x2 − 1) , ! y = φ2 (x) = − 3(x2 − 1) , ! y = φ3 (x) = 3(x2 − 1) , ! y = φ4 (x) = − 3(x2 − 1) , y 4 1 < x < ∞, 2 1 < x < ∞, −∞ < x < −1, -4 -2 −∞ < x < −1. 2 4 x 2 4 x -2 -4 (c) Only y = φ3 (x) satisfies y(−2) = 3. 34. (a) Setting x = 2 and y = −4 in 3x2 − y 2 = c we get 12 − 16 = −4 = c, so the explicit solution is ! y = − 3x2 + 4 , −∞ < x < ∞. (b) Setting c = 0 we have y = (−∞, ∞). √ √ 3x and y = − 3x, both defined on y 4 2 -4 -2 -2 -4 11 1.2 Initial-Value Problems In Problems 35–38, we consider the points on the graphs with x-coordinates x0 = −1, x0 = 0, and x0 = 1. The slopes of the tangent lines at these points are compared with the slopes given by y ′ (x0 ) in (a) through (f). 35. The graph satisfies the conditions in (b) and (f). 36. The graph satisfies the conditions in (e). 37. The graph satisfies the conditions in (c) and (d). 38. The graph satisfies the conditions in (a). 39. Integrating y ′ = 8e2x + 6x we obtain y= ) (8e2x + 6x)dx = 4e2x + 3x2 + c. Setting x = 0 and y = 9 we have 9 = 4 + c so c = 5 and y = 4e2x + 3x2 + 5. 40. Integrating y ′′ = 12x − 2 we obtain y′ = ) (12x − 2)dx = 6x2 − 2x + c1 . Then, integrating y ′ we obtain y= ) (6x2 − 2x + c1 )dx = 2x3 − x2 + c1 x + c2 . At x = 1 the y-coordinate of the point of tangency is y = −1 + 5 = 4. This gives the initial condition y(1) = 4. The slope of the tangent line at x = 1 is y ′ (1) = −1. From the initial conditions we obtain and 2 − 1 + c1 + c2 = 4 or c1 + c2 = 3 6 − 2 + c1 = −1 or c1 = −5. Thus, c1 = −5 and c2 = 8, so y = 2x3 − x2 − 5x + 8. 41. When x = 0 and y = or the black curve. 1 2 , y ′ = −1, so the only plausible solution curve is the one with negative slope at (0, 12 ), 42. If the solution is tangent to the x-axis at (x0 , 0), then y ′ = 0 when x = x0 and y = 0. Substituting these values into y ′ + 2y = 3x − 6 we get 0 + 0 = 3x0 − 6 or x0 = 2. 43. The theorem guarantees a unique (meaning single) solution through any point. Thus, there cannot be two distinct solutions through any point. 44. When y = 1 4 16 x , 1 y ′ = 41 x3 = x( 41 x2 ) = xy 1/2 , and y(2) = 16 (16) = 1. When * 0, x<0 y= 1 4 16 x , x ≥ 0 we have y′ = and y(2) = 1 16 (16) * 0, 1 3 4x , x<0 =x x≥0 * 0, 1 2 4x , x<0 = xy 1/2 , x≥0 = 1. The two different solutions are the same on the interval (0, ∞), which is all that is required by Theorem 1.1. 45. At t = 0, dP/dt = 0.15P (0) + 20 = 0.15(100) + 20 = 35. Thus, the population is increasing at a rate of 3,500 individuals per year. 12 1.3 Differential Equations as Mathematical Models If the population is 500 at time t = T then $ dP $$ = 0.15P (T ) + 20 = 0.15(500) + 20 = 95. $ dt $ t=T Thus, at this time, the population is increasing at a rate of 9,500 individuals per year. EXERCISES 1.3 Differential Equations as Mathematical Models dP dP = kP + r; = kP − r dt dt 2. Let b be the rate of births and d the rate of deaths. Then b = k1 P and d = k2 P . Since dP/dt = b − d, the 1. differential equation is dP/dt = k1 P − k2 P . 3. Let b be the rate of births and d the rate of deaths. Then b = k1 P and d = k2 P 2 . Since dP/dt = b − d, the differential equation is dP/dt = k1 P − k2 P 2 . dP 4. = k1 P − k2 P 2 − h, h > 0 dt 5. From the graph in the text we estimate T0 = 180◦ and Tm = 75◦ . We observe that when T = 85, dT /dt ≈ −1. From the differential equation we then have k= dT /dt −1 = = −0.1. T − Tm 85 − 75 6. By inspecting the graph in the text we take Tm to be Tm (t) = 80 − 30 cos πt/12. Then the temperature of the body at time t is determined by the differential equation . , π -/ dT = k T − 80 − 30 cos t , dt 12 t > 0. 7. The number of students with the flu is x and the number not infected is 1000 − x, so dx/dt = kx(1000 − x). 8. By analogy, with the differential equation modeling the spread of a disease, we assume that the rate at which the technological innovation is adopted is proportional to the number of people who have adopted the innovation and also to the number of people, y(t), who have not yet adopted it. If one person who has adopted the innovation is introduced into the population, then x + y = n + 1 and dx = kx(n + 1 − x), dt x(0) = 1. 9. The rate at which salt is leaving the tank is Rout (3 gal/min) · " # A A lb/gal = lb/min. 300 100 Thus dA/dt = A/100. The initial amount is A(0) = 50. 10. The rate at which salt is entering the tank is Rin = (3 gal/min) · (2 lb/gal) = 6 lb/min. 13 1.3 Differential Equations as Mathematical Models Since the solution is pumped out at a slower rate, it is accumulating at the rate of (3 − 2)gal/min = 1 gal/min. After t minutes there are 300 + t gallons of brine in the tank. The rate at which salt is leaving is " # A 2A Rout = (2 gal/min) · lb/gal = lb/min. 300 + t 300 + t The differential equation is 2A dA =6− . dt 300 + t 11. The rate at which salt is entering the tank is Rin = (3 gal/min) · (2 lb/gal) = 6 lb/min. Since the tank loses liquid at the net rate of 3 gal/min − 3.5 gal/min = −0.5 gal/min, after t minutes the number of gallons of brine in the tank is 300 − 12 t gallons. Thus the rate at which salt is leaving is # " 3.5A 7A A lb/gal · (3.5 gal/min) = lb/min = lb/min. Rout = 300 − t/2 300 − t/2 600 − t The differential equation is 7A dA =6− dt 600 − t dA 7 + A = 6. dt 600 − t or 12. The rate at which salt is entering the tank is Rin = (cin lb/gal) · (rin gal/min) = cin rin lb/min. Now let A(t) denote the number of pounds of salt and N (t) the number of gallons of brine in the tank at time t. The concentration of salt in the tank as well as in the outflow is c(t) = x(t)/N (t). But the number of gallons of brine in the tank remains steady, is increased, or is decreased depending on whether rin = rout , rin > rout , or rin < rout . In any case, the number of gallons of brine in the tank at time t is N (t) = N0 + (rin − rout )t. The output rate of salt is then " # A A Rout = lb/gal · (rout gal/min) = rout lb/min. N0 + (rin − rout )t N0 + (rin − rout )t The differential equation for the amount of salt, dA/dt = Rin − Rout , is dA A = cin rin − rout dt N0 + (rin − rout )t or dA rout + A = cin rin . dt N0 + (rin − rout )t 13. The volume of water in the tank at time t is V = Aw h. The differential equation is then ! 1 dV 1 , dh cAh ! = = −cAh 2gh = − 2gh . dt Aw dt Aw Aw " #2 π 2 = , Aw = 102 = 100, and g = 32, this becomes Using Ah = π 12 36 cπ/36 √ cπ √ dh =− 64h = − h. dt 100 450 14. The volume of water in the tank at time t is V = 13 πr2 h where r is the radius of the tank at height h. From % &2 4 the figure in the text we see that r/h = 8/20 so that r = 52 h and V = 13 π 52 h h = 75 πh3 . Differentiating with 4 respect to t we have dV /dt = 25 πh2 dh/dt or 25 dV dh = . dt 4πh2 dt 14 1.3 Differential Equations as Mathematical Models % 2 &2 √ From Problem 13 we have dV /dt = −cAh 2gh where c = 0.6, Ah = π 12 , and g = 32. Thus dV /dt = √ −2π h/15 and 0 √ 1 2π h 5 25 dh − = − 3/2 . = dt 4πh2 15 6h 15. Since i = dq/dt and L d2 q/dt2 + R dq/dt = E(t), we obtain L di/dt + Ri = E(t). 1 dq 16. By Kirchhoff’s second law we obtain R + q = E(t). dt C dv 17. From Newton’s second law we obtain m = −kv 2 + mg. dt 18. Since the barrel in Figure 1.35(b) in the text is submerged an additional y feet below its equilibrium position the number of cubic feet in the additional submerged portion is the volume of the circular cylinder: π×(radius)2 ×height or π(s/2)2 y. Then we have from Archimedes’ principle upward force of water on barrel = weight of water displaced = (62.4) × (volume of water displaced) = (62.4)π(s/2)2 y = 15.6πs2 y. It then follows from Newton’s second law that w d2 y = −15.6πs2 y g dt2 d2 y 15.6πs2 g + y = 0, dt2 w or where g = 32 and w is the weight of the barrel in pounds. 19. The net force acting on the mass is F = ma = m d2 x = −k(s + x) + mg = −kx + mg − ks. dt2 Since the condition of equilibrium is mg = ks, the differential equation is m d2 x = −kx. dt2 20. From Problem 19, without a damping force, the differential equation is m d2 x/dt2 = −kx. With a damping force proportional to velocity, the differential equation becomes m d2 x dx = −kx − β 2 dt dt or m d2 x dx +β + kx = 0. 2 dt dt 21. Let x(t) denote the height of the top of the chain at time t with the positive direction upward. The weight of the portion of chain off the ground is W = (x ft) · (1 lb/ft) = x. The mass of the chain is m = W/g = x/32. The net force is F = 5 − W = 5 − x. By Newton’s second law, dv dx d,x v =5−x or x +v = 160 − 32x. dt 32 dt dt Thus, the differential equation is x d2 x , dx -2 + 32x = 160. + dt2 dt 22. The force is the weight of the chain, 2L, so by Newton’s second law, of chain off the ground is m = 2(L − x)/g, we have d . 2(L − x) / v = 2L or dt g (L − x) 15 d [mv] = 2L. Since the mass of the portion dt , dx dv = Lg. +v − dt dt 1.3 Differential Equations as Mathematical Models Thus, the differential equation is (L − x) d2 x , dx -2 = Lg. − dt2 dt 23. From g = k/R2 we find k = gR2 . Using a = d2 r/dt2 and the fact that the positive direction is upward we get k gR2 d2 r = −a = − = − dt2 r2 r2 or d2 r gR2 + 2 = 0. dt2 r 24. The gravitational force on m is F = −kMr m/r2 . Since Mr = 4πδr3 /3 and M = 4πδR3 /3 we have Mr = r3 M/R3 and r3 M m/R3 mM Mr m = −k 3 r. F = −k 2 = −k r r2 R 2 2 Now from F = ma = d r/dt we have m d2 r mM = −k 3 r 2 dt R d2 r kM = − 3 r. 2 dt R or dA = k(M − A). dt dA 26. The differential equation is = k1 (M − A) − k2 A. dt 27. The differential equation is x′ (t) = r − kx(t) where k > 0. 25. The differential equation is 28. By the Pythagorean Theorem the slope of the tangent line is y ′ = ! −y s2 − y 2 . 29. We see from the figure that 2θ + α = π. Thus y y 2 tan θ . = tan α = tan(π − 2θ) = − tan 2θ = − −x 1 − tan2 θ Since the slope of the tangent line is y ′ = tan θ we have y/x = 2y ′ [1 − (y ′ )2 ] or y − y(y ′ )2 = 2xy ′ , which is the quadratic equation y(y ′ )2 + 2xy ′ − y = 0 in y ′ . Using the quadratic formula, we get ! ! −2x ± 4x2 + 4y 2 −x ± x2 + y 2 ′ y = = . 2y y Since dy/dx > 0, the differential equation is ! dy −x + x2 + y 2 = dx y or y (x,y) θ θα θ x α y φ x dy ! 2 − x + y 2 + x = 0. dx 30. The differential equation is dP/dt = kP , so from Problem 37 in Exercises 1.1, P = ekt , and a one-parameter family of solutions is P = cekt . 31. The differential equation in (3) is dT /dt = k(T − Tm ). When the body is cooling, T > Tm , so T − Tm > 0. Since T is decreasing, dT /dt < 0 and k < 0. When the body is warming, T < Tm , so T − Tm < 0. Since T is increasing, dT /dt > 0 and k < 0. 32. The differential equation in (8) is dA/dt = 6 − A/100. If A(t) attains a maximum, then dA/dt = 0 at this time and A = 600. If A(t) continues to increase without reaching a maximum, then A′ (t) > 0 for t > 0 and A cannot exceed 600. In this case, if A′ (t) approaches 0 as t increases to infinity, we see that A(t) approaches 600 as t increases to infinity. 33. This differential equation could describe a population that undergoes periodic fluctuations. 16 1.3 Differential Equations as Mathematical Models 34. (a) As shown in Figure 1.43(b) in the text, the resultant of the reaction force of magnitude F and the weight of magnitude mg of the particle is the centripetal force of magnitude mω 2 x. The centripetal force points to the center of the circle of radius x on which the particle rotates about the y-axis. Comparing parts of similar triangles gives F cos θ = mg and F sin θ = mω 2 x. (b) Using the equations in part (a) we find tan θ = F sin θ mω 2 x ω2 x = = F cos θ mg g or dy ω2 x = . dx g 35. From Problem 23, d2 r/dt2 = −gR2 /r2 . Since R is a constant, if r = R + s, then d2 r/dt2 = d2 s/dt2 and, using a Taylor series, we get d2 s R2 2gs = −g = −gR2 (R + s)−2 ≈ −gR2 [R−2 − 2sR−3 + · · · ] = −g + 3 + · · · . 2 dt (R + s)2 R Thus, for R much larger than s, the differential equation is approximated by d2 s/dt2 = −g. 36. (a) If ρ is the mass density of the raindrop, then m = ρV and , dm dV d . 4 3/ dr dr =ρ =ρ πr = ρ 4πr2 = ρS . dt dt dt 3 dt dt If dr/dt is a constant, then dm/dt = kS where ρ dr/dt = k or dr/dt = k/ρ. Since the radius is decreasing, k < 0. Solving dr/dt = k/ρ we get r = (k/ρ)t + c0 . Since r(0) = r0 , c0 = r0 and r = kt/ρ + r0 . d (b) From Newton’s second law, [mv] = mg, where v is the velocity of the raindrop. Then dt ,4 ,4 - dv dv dm m +v = mg or ρ πr3 + v(k4πr2 ) = ρ πr3 g. dt dt 3 dt 3 Dividing by 4ρπr3 /3 we get dv 3k + v=g dt ρr dv 3k/ρ v = g, k < 0. + dt kt/ρ + r0 or 37. We assume that the plow clears snow at a constant rate of k cubic miles per hour. Let t be the time in hours after noon, x(t) the depth in miles of the snow at time t, and y(t) the distance the plow has moved in t hours. Then dy/dt is the velocity of the plow and the assumption gives wx dy = k, dt where w is the width of the plow. Each side of this equation simply represents the volume of snow plowed in one hour. Now let t0 be the number of hours before noon when it started snowing and let s be the constant rate in miles per hour at which x increases. Then for t > −t0 , x = s(t + t0 ). The differential equation then becomes dy k 1 . = dt ws t + t0 Integrating, we obtain k [ ln(t + t0 ) + c ] ws where c is a constant. Now when t = 0, y = 0 so c = − ln t0 and " # t k ln 1 + . y= ws t0 y= 17 1.3 Differential Equations as Mathematical Models Finally, from the fact that when t = 1, y = 2 and when t = 2, y = 3, we obtain " #2 " #3 2 1 1+ = 1+ . t0 t0 Expanding and simplifying gives t20 + t0 − 1 = 0. Since t0 > 0, we find t0 ≈ 0.618 hours ≈ 37 minutes. Thus it started snowing at about 11:23 in the morning. dP dA 38. (1): = kP is linear (2): = kA is linear dt dt dx dT = k(T − Tm ) is linear (5): = kx(n + 1 − x) is nonlinear (3): dt dt dA A dX = k(α − X)(β − X) is nonlinear (8): =6− is linear (6): dt dt 100 dh dq d2 q Ah ! 1 (10): =− 2gh is nonlinear (11): L 2 + R + q = E(t) is linear dt Aw dt dt C d2 s dv (12): = −g is linear (14): m = mg − kv is linear 2 dt dt ds d2 x 64 d2 s = mg is linear (16): − x = 0 is linear (15): m 2 + k dt dt dt2 L (17): linearity or nonlinearity is determined by the manner in which W and T1 involve x. 39. At time t, when the population is 2 million cells, the differential equation P ′ (t) = 0.15P (t) gives the rate of increase at time t. Thus, when P (t) = 2 (million cells), the rate of increase is P ′ (t) = 0.15(2) = 0.3 million cells per hour or 300,000 cells per hour. 40. Setting A′ (t) = −0.002 and solving A′ (t) = −0.0004332A(t) for A(t), we obtain A(t) = A′ (t) −0.002 = ≈ 4.6 grams. −0.0004332 −0.0004332 CHAPTER 1 REVIEW EXERCISES d dy c1 ekx = c1 kekx ; = ky dx dx dy dy d (5 + c1 e−2x ) = −2c1 e−2x = −2(5 + c1 e−2x − 5); = −2(y − 5) or = −2y + 10 2. dx dx dx d (c1 cos kx + c2 sin kx) = −kc1 sin kx + kc2 cos kx; 3. dx d2 (c1 cos kx + c2 sin kx) = −k 2 c1 cos kx − k 2 c2 sin kx = −k 2 (c1 cos kx + c2 sin kx); dx2 d2 y d2 y 2 = −k y or + k2 y = 0 dx2 dx2 d 4. (c1 cosh kx + c2 sinh kx) = kc1 sinh kx + kc2 cosh kx; dx d2 (c1 cosh kx + c2 sinh kx) = k 2 c1 cosh kx + k 2 c2 sinh kx = k 2 (c1 cosh kx + c2 sinh kx); dx2 1. 18 CHAPTER 1 REVIEW EXERCISES d2 y = k2 y dx2 d2 y − k2 y = 0 dx2 or 5. y = c1 ex + c2 xex ; y ′ = c1 ex + c2 xex + c2 ex ; y ′′ = c1 ex + c2 xex + 2c2 ex ; y ′′ + y = 2(c1 ex + c2 xex ) + 2c2 ex = 2(c1 ex + c2 xex + c2 ex ) = 2y ′ ; y ′′ − 2y ′ + y = 0 6. y ′ = −c1 ex sin x + c1 ex cos x + c2 ex cos x + c2 ex sin x; y ′′ = −c1 ex cos x − c1 ex sin x − c1 ex sin x + c1 ex cos x − c2 ex sin x + c2 ex cos x + c2 ex cos x + c2 ex sin x = −2c1 ex sin x + 2c2 ex cos x; y ′′ − 2y ′ = −2c1 ex cos x − 2c2 ex sin x = −2y; 7. a,d 8. c y ′′ − 2y ′ + 2y = 0 10. a,c 9. b 11. b 12. a,b,d 13. A few solutions are y = 0, y = c, and y = ex . 14. Easy solutions to see are y = 0 and y = 3. 15. The slope of the tangent line at (x, y) is y ′ , so the differential equation is y ′ = x2 + y 2 . 16. The rate at which the slope changes is dy ′ /dx = y ′′ , so the differential equation is y ′′ = −y ′ or y ′′ + y ′ = 0. 17. (a) The domain is all real numbers. (b) Since y ′ = 2/3x1/3 , the solution y = x2/3 is undefined at x = 0. This function is a solution of the differential equation on (−∞, 0) and also on (0, ∞). 18. (a) Differentiating y 2 − 2y = x2 − x + c we obtain 2yy ′ − 2y ′ = 2x − 1 or (2y − 2)y ′ = 2x − 1. (b) Setting x = 0 and y = 1 in the solution we have 1 − 2 = 0 − 0 + c or c = −1. Thus, a solution of the initial-value problem is y 2 − 2y = x2 − x − 1. ! (c) Solving y 2 − 2y − (x2 − x − 1) = 0 by the quadratic formula we get y = (2 ± 4 + 4(x2 − x − 1) )/2 ! ! √ = 1± x2 − x = 1± x(x − 1) . Since x(x−1) ≥ 0 for x ≤ 0 or x ≥ 1, we see that neither y = 1+ x(x − 1) ! nor y = 1 − x(x − 1) is differentiable at x = 0. Thus, both functions are solutions of the differential equation, but neither is a solution of the initial-value problem. 19. Setting x = x0 and y = 1 in y = −2/x + x, we get 1=− 2 + x0 x0 or x20 − x0 − 2 = (x0 − 2)(x0 + 1) = 0. Thus, x0 = 2 or x0 = −1. Since x = 0 in y = −2/x+x, we see that y = −2/x+x is a solution of the initial-value problem xy ′ + y = 2x, y(−1) = 1, on the interval (−∞, 0) and y = −2/x + x is a solution of the initial-value problem xy ′ + y = 2x, y(2) = 1, on the interval (0, ∞). 20. From the differential equation, y ′ (1) = 12 + [y(1)]2 = 1 + (−1)2 = 2 > 0, so y(x) is increasing in some neighborhood of x = 1. From y ′′ = 2x + 2yy ′ we have y ′′ (1) = 2(1) + 2(−1)(2) = −2 < 0, so y(x) is concave down in some neighborhood of x = 1. 21. (a) y 3 y 3 2 2 1 1 -3 -2 -1 -1 1 2 3 x -3 -2 -1 -1 -2 -2 -3 -3 y = x2 + c1 1 2 y = −x2 + c2 19 3 x CHAPTER 1 REVIEW EXERCISES (b) When y = x2 + c1 , y ′ = 2x and (y ′ )2 = 4x2 . When y = −x2 + c2 , y ′ = −2x and (y ′ )2 = 4x2 . * −x2 , x ≤ 0 2 2 (c) Pasting together x , x ≥ 0, and −x , x ≤ 0, we get y = x2 , x > 0. $ √ 22. The slope of the tangent line is y ′ $(−1,4) = 6 4 + 5(−1)3 = 7. 23. Differentiating y = x sin x + x cos x we get y ′ = x cos x + sin x − x sin x + cos x and y ′′ = −x sin x + cos x + cos x − x cos x − sin x − sin x = −x sin x − x cos x + 2 cos x − 2 sin x. Thus y ′′ + y = −x sin x − x cos x + 2 cos x − 2 sin x + x sin x + x cos x = 2 cos x − 2 sin x. An interval of definition for the solution is (−∞, ∞). 24. Differentiating y = x sin x + (cos x) ln(cos x) we get ′ y = x cos x + sin x + cos x " − sin x cos x # − (sin x) ln(cos x) = x cos x + sin x − sin x − (sin x) ln(cos x) = x cos x − (sin x) ln(cos x) and y ′′ = −x sin x + cos x − sin x " − sin x cos x # − (cos x) ln(cos x) sin2 x − (cos x) ln(cos x) cos x 1 − cos2 x − (cos x) ln(cos x) = −x sin x + cos x + cos x = −x sin x + cos x + sec x − cos x − (cos x) ln(cos x) = −x sin x + cos x + = −x sin x + sec x − (cos x) ln(cos x). Thus y ′′ + y = −x sin x + sec x − (cos x) ln(cos x) + x sin x + (cos x) ln(cos x) = sec x. To obtain an interval of definition we note that the domain of ln x is (0, ∞), so we must have cos x > 0. Thus, an interval of definition is (−π/2, π/2). 25. Differentiating y = sin(ln x) we obtain y ′ = cos(ln x)/x and y ′′ = −[sin(ln x) + cos(ln x)]/x2 . Then " # sin(ln x) + cos(ln x) cos(ln x) x2 y ′′ + xy ′ + y = x2 − + sin(ln x) = 0. +x 2 x x An interval of definition for the solution is (0, ∞). 26. Differentiating y = cos(ln x) ln(cos(ln x)) + (ln x) sin(ln x) we obtain " # " # sin(ln x) sin(ln x) cos(ln x) sin(ln x) 1 y ′ = cos(ln x) + − + ln(cos(ln x)) − + ln x cos(ln x) x x x x =− ln(cos(ln x)) sin(ln x) (ln x) cos(ln x) + x x 20 CHAPTER 1 REVIEW EXERCISES and ' , sin(ln x) -( 1 cos(ln x) 1 y = −x ln(cos(ln x)) + sin(ln x) − x cos(ln x) x x2 ' , sin(ln x) - cos(ln x) ( 1 1 1 + − (ln x) cos(ln x) 2 + ln(cos(ln x)) sin(ln x) 2 + x (ln x) − x x x x2 x ' 1 sin2 (ln x) = 2 − ln(cos(ln x)) cos(ln x) + + ln(cos(ln x)) sin(ln x) x cos(ln x) ( − (ln x) sin(ln x) + cos(ln x) − (ln x) cos(ln x) . ′′ Then x2 y ′′ + xy ′ + y = − ln(cos(ln x)) cos(ln x) + sin2 (ln x) + ln(cos(ln x)) sin(ln x) − (ln x) sin(ln x) cos(ln x) + cos(ln x) − (ln x) cos(ln x) − ln(cos(ln x)) sin(ln x) + (ln x) cos(ln x) + cos(ln x) ln(cos(ln x)) + (ln x) sin(ln x) sin (ln x) sin2 (ln x) + cos2 (ln x) 1 + cos(ln x) = = = sec(ln x). cos(ln x) cos(ln x) cos(ln x) 2 = To obtain an interval of definition, we note that the domain of ln x is (0, ∞), so we must have cos(ln x) > 0. Since cos x > 0 when −π/2 < x < π/2, we require −π/2 < ln x < π/2. Since ex is an increasing function, this is equivalent to e−π/2 < x < eπ/2 . Thus, an interval of definition is (e−π/2 , eπ/2 ). (Much of this problem is more easily done using a computer algebra system such as Mathematica or Maple.) 27. From the graph we see that estimates for y0 and y1 are y0 = −3 and y1 = 0. 28. The differential equation is dh cA0 ! =− 2gh . dt Aw Using A0 = π(1/24)2 = π/576, Aw = π(2)2 = 4π, and g = 32, this becomes c √ dh cπ/576 √ 64h = h. =− dt 4π 288 21 2 First-Order Differential Equations EXERCISES 2.1 Solution Curves Without the Solution y 3 1. 2. y 10 2 5 1 -3 -2 -1 1 2 x 0 x 3 -1 -5 -2 -10 -3 3. y -5 10 y 4. 4 5 0 4 2 2 x 0 x 0 -2 -2 -4 -4 5. -2 0 2 4 -4 y 6. 4 2 0 2 4 y 4 2 x 0 x 0 -2 -4 -2 -2 -2 0 2 4 -4 22 -2 0 2 4 2.1 7. y 8. 4 x 0 -2 -4 -2 0 2 4 -4 y 10. 4 2 x 0 2 4 y 4 x 0 -2 -2 -4 -2 0 2 4 -4 y 12. 4 2 -2 0 2 4 y 4 2 x 0 x 0 -2 -2 -4 13. -2 2 0 11. x 0 -2 9. y 4 2 2 -4 Solution Curves Without the Solution -2 0 2 4 -4 y -2 2 4 y 14. 3 0 4 2 2 1 x 0 x 0 -1 -2 -2 -4 -3 -3 -2 -1 0 1 2 3 -4 23 -2 0 2 4 2.1 Solution Curves Without the Solution 15. (a) The isoclines have the form y = −x + c, which are straight y 3 lines with slope −1. 2 1 -3 -2 -1 1 3 x 2 -1 -2 -3 (b) The isoclines have the form x2 + y 2 = c, which are circles y centered at the origin. 2 1 -2 -1 1 x 2 -1 -2 16. (a) When x = 0 or y = 4, dy/dx = −2 so the lineal elements have slope −2. When y = 3 or y = 5, dy/dx = x−2, so the lineal elements at (x, 3) and (x, 5) have slopes x − 2. (b) At (0, y0 ) the solution curve is headed down. If y → ∞ as x increases, the graph must eventually turn around and head up, but while heading up it can never cross y = 4 where a tangent line to a solution curve must have slope −2. Thus, y cannot approach ∞ as x approaches ∞. 1 2 2x , y ′ = x2 − 2y is positive and the portions of solution curves “outside” the nullcline parabola are increasing. When y > 21 x2 , y ′ = x2 − 2y is negative and the portions of the solution curves “inside” the 17. When y < nullcline parabola are decreasing. y 3 2 1 x 0 -1 -2 -3 -3 -2 -1 0 1 2 3 18. (a) Any horizontal lineal element should be at a point on a nullcline. In Problem 1 the nullclines are x2 −y 2 = 0 or y = ±x. In Problem 3 the nullclines are 1 − xy = 0 or y = 1/x. In Problem 4 the nullclines are (sin x) cos y = 0 or x = nπ and y = π/2 + nπ, where n is an integer. The graphs on the next page show the nullclines for the differential equations in Problems 1, 3, and 4 superimposed on the corresponding direction field. 24 2.1 y Solution Curves Without the Solution y y 4 3 4 2 2 2 1 x 0 x 0 -1 -2 -2 -2 x 0 -4 -3 -3 -2 -1 0 1 2 Problem 1 -4 -4 3 4 -2 0 2 Problem 3 -4 -2 0 2 Problem 4 4 (b) An autonomous first-order differential equation has the form y ′ = f (y). Nullclines have the form y = c where f (c) = 0. These are the graphs of the equilibrium solutions of the differential equation. 19. Writing the differential equation in the form dy/dx = y(1 − y)(1 + y) we see that critical points are located at y = −1, y = 0, and y = 1. The phase portrait is shown at the right. y (a) 1 y (b) 5 4 0 1 3 -1 2 1 -2 1 2 y (c) -1 1 (d) -1 1 2 x y 1 -2 2 x x 2 x -1 -2 -3 -4 -1 -5 20. Writing the differential equation in the form dy/dx = y 2 (1 − y)(1 + y) we see that critical points are located at y = −1, y = 0, and y = 1. The phase portrait is shown at the right. y (a) 1 y (b) 5 4 0 1 3 2 -1 1 -2 1 y 2 -1 1 2 x y (c) (d) -2 -2 -1 1 2 x x -1 -1 -2 -3 -4 -1 -5 25 x 2.1 Solution Curves Without the Solution 21. Solving y 2 − 3y = y(y − 3) = 0 we obtain the critical points 0 and 3. From the phase portrait we see that 0 is asymptotically stable (attractor) and 3 is unstable (repeller). 3 0 22. Solving y 2 − y 3 = y 2 (1 − y) = 0 we obtain the critical points 0 and 1. From the phase portrait we see that 1 is asymptotically stable (attractor) and 0 is semi-stable. 1 0 23. Solving (y − 2)4 = 0 we obtain the critical point 2. From the phase portrait we see that 2 is semi-stable. 2 24. Solving 10 + 3y − y 2 = (5 − y)(2 + y) = 0 we obtain the critical points −2 and 5. From the phase portrait we see that 5 is asymptotically stable (attractor) and −2 is unstable (repeller). 5 -2 26 2.1 Solution Curves Without the Solution 25. Solving y 2 (4 − y 2 ) = y 2 (2 − y)(2 + y) = 0 we obtain the critical points −2, 0, and 2. From the phase portrait we see that 2 is asymptotically stable (attractor), 0 is semi-stable, and −2 is unstable (repeller). 2 0 -2 26. Solving y(2 − y)(4 − y) = 0 we obtain the critical points 0, 2, and 4. From the phase portrait we see that 2 is asymptotically stable (attractor) and 0 and 4 are unstable (repellers). 4 2 0 27. Solving y ln(y + 2) = 0 we obtain the critical points −1 and 0. From the phase portrait we see that −1 is asymptotically stable (attractor) and 0 is unstable (repeller). 0 -1 -2 28. Solving yey − 9y = y(ey − 9) = 0 we obtain the critical points 0 and ln 9. From the phase portrait we see that 0 is asymptotically stable (attractor) and ln 9 is unstable (repeller). ln 9 0 29. The critical points are 0 and c because the graph of f (y) is 0 at these points. Since f (y) > 0 for y < 0 and y > c, the graph of the solution is increasing on (−∞, 0) and (c, ∞). Since f (y) < 0 for 0 < y < c, the graph of the solution is decreasing on (0, c). 27 2.1 Solution Curves Without the Solution y c c x 0 30. The critical points are approximately at −2, 2, 0.5, and 1.7. Since f (y) > 0 for y < −2.2 and 0.5 < y < 1.7, the graph of the solution is increasing on (−∞, −2.2) and (0.5, 1.7). Since f (y) < 0 for −2.2 < y < 0.5 and y > 1.7, the graph is decreasing on (−2.2, 0.5) and (1.7, ∞). y 2 1.7 1 0.5 -2 -1 1 2 x -1 -2 -2.2 31. From the graphs of z = π/2 and z = sin y we see that (π/2)y − sin y = 0 has only three solutions. By inspection we see that the critical points are −π/2, 0, and π/2. 1 !Π From the graph at the right we see that y Π -1 2 y − sin y π 2 y − sin y π Π #### 2 Π ! #### 2 ! ! < 0 for y < −π/2 > 0 for y > π/2 > 0 for < 0 for − π/2 < y < 0 0 < y < π/2. Π #### 2 0 Π ! #### 2 This enables us to construct the phase portrait shown at the right. From this portrait we see that π/2 and −π/2 are unstable (repellers), and 0 is asymptotically stable (attractor). 32. For dy/dx = 0 every real number is a critical point, and hence all critical points are nonisolated. 33. Recall that for dy/dx = f (y) we are assuming that f and f ′ are continuous functions of y on some interval I. Now suppose that the graph of a nonconstant solution of the differential equation crosses the line y = c. If the point of intersection is taken as an initial condition we have two distinct solutions of the initial-value problem. This violates uniqueness, so the graph of any nonconstant solution must lie entirely on one side of any equilibrium solution. Since f is continuous it can only change signs at a point where it is 0. But this is a critical point. Thus, f (y) is completely positive or completely negative in each region Ri . If y(x) is oscillatory 28 2.1 Solution Curves Without the Solution or has a relative extremum, then it must have a horizontal tangent line at some point (x0 , y0 ). In this case y0 would be a critical point of the differential equation, but we saw above that the graph of a nonconstant solution cannot intersect the graph of the equilibrium solution y = y0 . 34. By Problem 33, a solution y(x) of dy/dx = f (y) cannot have relative extrema and hence must be monotone. Since y ′ (x) = f (y) > 0, y(x) is monotone increasing, and since y(x) is bounded above by c2 , limx→∞ y(x) = L, where L ≤ c2 . We want to show that L = c2 . Since L is a horizontal asymptote of y(x), limx→∞ y ′ (x) = 0. Using the fact that f (y) is continuous we have f (L) = f ( lim y(x)) = lim f (y(x)) = lim y ′ (x) = 0. x→∞ x→∞ x→∞ But then L is a critical point of f . Since c1 < L ≤ c2 , and f has no critical points between c1 and c2 , L = c2 . 35. Assuming the existence of the second derivative, points of inflection of y(x) occur where y ′′ (x) = 0. From dy/dx = f (y) we have d2 y/dx2 = f ′ (y) dy/dx. Thus, the y-coordinate of a point of inflection can be located by solving f ′ (y) = 0. (Points where dy/dx = 0 correspond to constant solutions of the differential equation.) 36. Solving y 2 − y − 6 = (y − 3)(y + 2) = 0 we see that 3 and −2 are critical points. 2 y 5 2 Now d y/dx = (2y −1) dy/dx = (2y −1)(y −3)(y +2), so the only possible point of inflection is at y = 12 , although the concavity of solutions can be different on either side of y = −2 and y = 3. Since y ′′ (x) < 0 for y < −2 and ′′ and y (x) > 0 for −2 < concave down for y < −2 y < 21 and y > 3, and 21 < y < 3 and 1 2 < y < 3, we see that solution curves are concave up for −2 < y < 12 and -5 y > 3. Points of inflection of solutions of autonomous differential equations will have the same y-coordinates because between critical points they are horizontal translates of each other. 5 x -5 37. If (1) in the text has no critical points it has no constant solutions. The solutions have neither an upper nor lower bound. Since solutions are monotonic, every solution assumes all real values. 38. The critical points are 0 and b/a. From the phase portrait we see that 0 is an attractor and b/a is a repeller. Thus, if an initial population satisfies P0 > b/a, the population becomes unbounded as t increases, most probably in finite time, i.e. P (t) → ∞ as t → T . If 0 < P0 < b/a, then the population eventually dies out, that is, P (t) → 0 as t → ∞. Since population P > 0 we do not consider the case P0 < 0. 39. (a) Writing the differential equation in the form # dv k " mg = −v dt m k we see that a critical point is mg/k. From the phase portrait we see that mg/k is an asymptotically stable critical point. Thus, limt→∞ v = mg/k. 29 b #### a 0 mg ####### k 2.1 Solution Curves Without the Solution (b) Writing the differential equation in the form $% & $% & # dv k " mg k mg mg 2 = −v = −v +v dt m k m k k ' we see that the only physically meaningful critical point is mg/k. ' From the phase portrait we see that mg/k is an asymptotically stable ' critical point. Thus, limt→∞ v = mg/k. mg !"""""""" #######" k 40. (a) From the phase portrait we see that critical points are α and β. Let X(0) = X0 . If X0 < α, we see that X → α as t → ∞. If α < X0 < β, we see that X → α as t → ∞. If X0 > β, we see that X(t) increases in an unbounded manner, but more specific behavior of X(t) as t → ∞ is not known. Β Α (b) When α = β the phase portrait is as shown. If X0 < α, then X(t) → α as t → ∞. If X0 > α, then X(t) increases in an unbounded manner. This could happen in a finite amount of time. That is, the phase portrait does not indicate that X becomes unbounded as t → ∞. Α (c) When k = 1 and α = β the differential equation is dX/dt = (α − X)2 . For X(t) = α − 1/(t + c) we have dX/dt = 1/(t + c)2 and ( $ &)2 1 dX 1 2 (α − X) = α − α − = . = 2 t+c (t + c) dt For X(0) = α/2 we obtain X(t) = α − 1 . t + 2/α X(t) = α − 1 . t − 1/α For X(0) = 2α we obtain 30 2.2 X X 2α α α −2/α Separable Variables α/2 t 1/α t For X0 > α, X(t) increases without bound up to t = 1/α. For t > 1/α, X(t) increases but X → α as t→∞ EXERCISES 2.2 Separable Variables In many of the following problems we will encounter an expression of the form ln |g(y)| = f (x) + c. To solve for g(y) we exponentiate both sides of the equation. This yields |g(y)| = ef (x)+c = ec ef (x) which implies g(y) = ±ec ef (x) . Letting c1 = ±ec we obtain g(y) = c1 ef (x) . 1. From dy = sin 5x dx we obtain y = − 15 cos 5x + c. 2. From dy = (x + 1)2 dx we obtain y = 13 (x + 1)3 + c. 3. From dy = −e−3x dx we obtain y = 31 e−3x + c. 1 1 1 dy = dx we obtain − = x + c or y = 1 − . (y − 1)2 y−1 x+c 1 4 5. From dy = dx we obtain ln |y| = 4 ln |x| + c or y = c1 x4 . y x 1 1 1 . 6. From 2 dy = −2x dx we obtain − = −x2 + c or y = 2 y y x + c1 4. From 7. From e−2y dy = e3x dx we obtain 3e−2y + 2e3x = c. + * 1 8. From yey dy = e−x + e−3x dx we obtain yey − ey + e−x + e−3x = c. 3 $ & 2 1 y x3 1 9. From y + 2 + dy = x2 ln x dx we obtain + 2y + ln |y| = ln |x| − x3 + c. y 2 3 9 1 2 1 1 dy = dx we obtain = + c. 10. From (2y + 3)2 (4x + 5)2 2y + 3 4x + 5 1 1 11. From dy = − 2 dx or sin y dy = − cos2 x dx = − 21 (1 + cos 2x) dx we obtain csc y sec x − cos y = − 12 x − 41 sin 2x + c or 4 cos y = 2x + sin 2x + c1 . 12. From 2y dy = − sin 3x dx or 2y dy = − tan 3x sec2 3x dx we obtain y 2 = − 16 sec2 3x + c. cos3 3x 31 2.2 Separable Variables 13. From 14. From ey 2 (ey + 1) y 1/2 (1 + y 2 ) −ex y = 21 (ex + 1) + c. 3 dx we obtain − (e + 1) (ex + 1) * +1/2 * +1/2 x dy = dx we obtain 1 + y 2 = 1 + x2 + c. 1/2 (1 + x2 ) dy = −1 −2 1 dS = k dr we obtain S = cekr . S 1 From dQ = k dt we obtain ln |Q − 70| = kt + c or Q − 70 = c1 ekt . Q − 70 , , $ & , P , 1 1 1 , , From dP = + dP = dt we obtain ln |P | − ln |1 − P | = t + c so that ln , 1 − P , = t + c or P − P2 P 1−P P c1 et = c1 et . Solving for P we have P = . 1−P 1 + c1 et + * 1 t+2 t+2 From dN = tet+2 − 1 dt we obtain ln |N | = tet+2 − et+2 − t + c or N = c1 ete −e −t . N $ & $ & y−2 x−1 5 5 From dy = dx or 1 − dy = 1 − dx we obtain y − 5 ln |y + 3| = x − 5 ln |x + 4| + c y+3 x+4 y+3 x+4 $ &5 x+4 = c1 ex−y . or y+3 $ & $ & y+1 x+2 2 5 From dy = dx or 1 + dy = 1 + dx we obtain y + 2 ln |y − 1| = x + 5 ln |x − 3| + c y−1 x−3 y−1 x−3 (y − 1)2 or = c1 ex−y . (x − 3)5 $ 2 & x 1 −1 1 2 dy we obtain 2 x = sin y + c or y = sin From x dx = ' + c1 . 2 1 − y2 15. From 16. 17. 18. 19. 20. 21. 22. From 1 1 ex 1 1 . dy = dx = dx we obtain − = tan−1 ex + c or y = − y2 ex + e−x (ex )2 + 1 y tan−1 ex + c 1 dx = 4 dt we obtain tan−1 x = 4t + c. Using x(π/4) = 1 we find c = −3π/4. The solution of the +1 $ & 3π 3π −1 initial-value problem is tan x = 4t − or x = tan 4t − . 4 4 $ & $ & 1 1 1 1 1 1 1 1 24. From 2 dy = dx we obtain dy = 2 dx or − − y −1 x −1 2 y−1 y+1 2 x−1 x+1 c(x − 1) y−1 ln |y − 1| − ln |y + 1| = ln |x − 1| − ln |x + 1| + ln c or = . Using y(2) = 2 we find y+1 x+1 23. From x2 c = 1. A solution of the initial-value problem is y−1 x−1 = or y = x. y+1 x+1 & $ 1 1 1 1 1−x 25. From dy = dx we obtain ln |y| = − − ln |x| = c or xy = c1 e−1/x . Using y(−1) = −1 dx = − 2 2 y x x x x we find c1 = e−1 . The solution of the initial-value problem is xy = e−1−1/x or y = e−(1+1/x) /x. 1 dy = dt we obtain − 21 ln |1 − 2y| = t + c or 1 − 2y = c1 e−2t . Using y(0) = 5/2 we find c1 = −4. 1 − 2y The solution of the initial-value problem is 1 − 2y = −4e−2t or y = 2e−2t + 12 . 26. From 27. Separating variables and integrating we obtain √ dx dy −' = 0 and 2 1−x 1 − y2 32 sin−1 x − sin−1 y = c. 2.2 Setting x = 0 and y = −1 sin 28. From x − sin −1 √ Separable Variables 3/2 we obtain c = −π/3. Thus, an implicit solution of the initial-value problem is y = π/3. Solving for y and using an addition formula from trigonometry, we get √ √ " π ' x 3 1 − x2 π π# = x cos + 1 − x2 sin = + . y = sin sin−1 x + 3 3 3 2 2 1 −x dy = 2 dx we obtain 2 1 + (2y) 1 + (x2 ) 1 1 tan−1 2y = − tan−1 x2 + c or 2 2 tan−1 2y + tan−1 x2 = c1 . Using y(1) = 0 we find c1 = π/4. Thus, an implicit solution of the initial-value problem is −1 −1 2 tan 2y + tan x = π/4 . Solving for y and using a trigonometric identity we get # "π 2y = tan − tan−1 x2 4 "π # 1 − tan−1 x2 y = tan 2 4 π 1 tan 4 − tan(tan−1 x2 ) = 2 1 + tan π4 tan(tan−1 x2 ) = 1 1 − x2 . 2 1 + x2 29. (a) The equilibrium solutions y(x) = 2 and y(x) = −2 satisfy the initial conditions y(0) = 2 and y(0) = −2, 1 4 respectively. Setting x = 1=2 and y = 1 in y = 2(1 + ce4x )/(1 − ce4x ) we obtain 1 + ce , 1 − ce 1 − ce = 2 + 2ce, −1 = 3ce, and c = − 1 . 3e The solution of the corresponding initial-value problem is y=2 1 − 31 e4x−1 3 − e4x−1 1 4x−1 = 2 3 + e4x−1 . 1 + 3e (b) Separating variables and integrating yields 1 1 ln |y − 2| − ln |y + 2| + ln c1 = x 4 4 ln |y − 2| − ln |y + 2| + ln c = 4x , c(y − 2) , , , ln , , = 4x y+2 y−2 = e4x . c y+2 Solving for y we get y = 2(c + e4x )/(c − e4x ). The initial condition y(0) = −2 implies 2(c + 1)/(c − 1) = −2 which yields c = 0 and y(x) = −2. The initial condition y(0) = 2 does not correspond to a value of c, and it must simply be recognized that y(x) = 2 is a solution of the initial-value problem. Setting x = 14 and y = 1 in y = 2(c + e4x )/(c − e4x ) leads to c = −3e. Thus, a solution of the initial-value problem is y=2 30. Separating variables, we have dy dx = y2 − y x 3 − e4x−1 −3e + e4x = 2 . −3e − e4x 3 + e4x−1 or - 33 dy = ln |x| + c. y(y − 1) 2.2 Separable Variables Using partial fractions, we obtain - $ 1 1 − y−1 y & dy = ln |x| + c ln |y − 1| − ln |y| = ln |x| + c , , ,y − 1, ,=c , ln , xy , y−1 = ec = c1 . xy Solving for y we get y = 1/(1 − c1 x). We note by inspection that y = 0 is a singular solution of the differential equation. (a) Setting x = 0 and y = 1 we have 1 = 1/(1 − 0), which is true for all values of c1 . Thus, solutions passing through (0, 1) are y = 1/(1 − c1 x). (b) Setting x = 0 and y = 0 in y = 1/(1 − c1 x) we get 0 = 1. Thus, the only solution passing through (0, 0) is y = 0. (c) Setting x = 1 2 and y = 1 2 we have 1 2 = 1/(1 − 1 2 c1 ), so c1 = −2 and y = 1/(1 + 2x). 1 4 we have 41 = 1/(1 − 2c1 ), so c1 = − 32 and y = 1/(1 + 23 x) = 2/(2 + 3x). ' 31. Singular solutions of dy/dx = x 1 − y 2 are y = −1 and y = 1. A singular solution of (d) Setting x = 2 and y = (ex + e−x )dy/dx = y 2 is y = 0. 32. Differentiating ln(x2 + 10) + csc y = c we get 2x dy − csc y cot y = 0, x2 + 10 dx or 2x 1 cos y dy − · = 0, x2 + 10 sin y sin y dx 2x sin2 y dx − (x2 + 10) cos y dy = 0. Writing the differential equation in the form dy 2x sin2 y = 2 dx (x + 10) cos y we see that singular solutions occur when sin2 y = 0, or y = kπ, where k is an integer. 33. The singular solution y = 1 satisfies the initial-value problem. y 1.01 1 -0.004-0.002 0.98 0.97 34 0.002 0.004 x 2.2 34. Separating variables we obtain − dy = dx. Then (y − 1)2 Separable Variables y 1.02 1 x+c−1 = x + c and y = . y−1 x+c 1.01 Setting x = 0 and y = 1.01 we obtain c = −100. The solution is y= x − 101 . x − 100 -0.004-0.002 0.002 0.004 x 0.99 0.98 dy = dx. Then 35. Separating variables we obtain (y − 1)2 + 0.01 y 1.0004 1 x+c tan . 10 10 Setting x = 0 and y = 1 we obtain c = 0. The solution is 10 tan−1 10(y − 1) = x + c and y = 1 + y =1+ 1.0002 x 1 tan . 10 10 -0.004-0.002 0.002 0.004 x 0.9998 0.9996 dy = dx. Then, from (11) in (y − 1)2 − 0.01 1 this section of the manual with u = y − 1 and a = 10 , we get , , , 10y − 11 , , = x + c. 5 ln ,, 10y − 9 , 36. Separating variables we obtain y 1.0004 1.0002 Setting x = 0 and y = 1 we obtain c = 5 ln 1 = 0. The solution is , , , 10y − 11 , , = x. 5 ln ,, 10y − 9 , -0.004-0.002 x 0.9998 Solving for y we obtain y= 0.002 0.004 0.9996 11 + 9ex/5 . 10 + 10ex/5 Alternatively, we can use the fact that dy y−1 1 =− tanh−1 = −10 tanh−1 10(y − 1). (y − 1)2 − 0.01 0.1 0.1 (We use the inverse hyperbolic tangent because |y − 1| < 0.1 or 0.9 < y < 1.1. This follows from the initial condition y(0) = 1.) Solving the above equation for y we get y = 1 + 0.1 tanh(x/10). 37. Separating variables, we have Integrating, we get dy dy = = 3 y−y y(1 − y)(1 + y) ln |y| − $ 1 1/2 1/2 + − y 1−y 1+y & 1 1 ln |1 − y| − ln |1 + y| = x + c. 2 2 35 dy = dx. 2.2 Separable Variables When y > 1, this becomes 1 y 1 = x + c. ln(y − 1) − ln(y + 1) = ln ' 2 2 y2 − 1 √ √ √ Letting x = 0 and y = 2 we find c = ln(2/ 3 ). Solving for y we get y1 (x) = 2ex / 4e2x − 3 , where x > ln( 3/2). ln y − When 0 < y < 1 we have 1 1 y = x + c. ln(1 − y) − ln(1 + y) = ln ' 2 2 1 − y2 √ √ we find c = ln(1/ 3 ). Solving for y we get y2 (x) = ex / e2x + 3 , where −∞ < x < ∞. ln y − 1 2 Letting x = 0 and y = When −1 < y < 0 we have 1 1 −y = x + c. ln(1 − y) − ln(1 + y) = ln ' 2 2 1 − y2 √ √ we find c = ln(1/ 3 ). Solving for y we get y3 (x) = −ex / e2x + 3 , where ln(−y) − Letting x = 0 and y = − 21 −∞ < x < ∞. When y < −1 we have 1 1 −y = x + c. ln(1 − y) − ln(−1 − y) = ln ' 2 2 y2 − 1 √ √ Letting x = 0 and y = −2 we find c = ln(2/ 3 ). Solving for y we get y4 (x) = −2ex / 4e2x − 3 , where √ x > ln( 3/2). ln(−y) − y y y y 4 4 4 4 2 2 2 2 1 2 3 4 5x -4 -2 2 4 x -4 -2 2 4 x 1 -2 -2 -2 -2 -4 -4 -4 -4 38. (a) The second derivative of y is d y dy/dx 1/(y − 3) 1 =− =− =− . 2 2 2 dx (y − 1) (y − 3) (y − 3)3 5x 4 6 2 The solution curve is concave down when d y/dx < 0 or y > 3, and concave up when d2 y/dx2 > 0 or y < 3. From the phase portrait we see that the solution curve is decreasing when y < 3 and increasing when y > 3. 3 y 8 2 2 2 4 3 2 -4 -2 2 x 4 -2 (b) Separating variables and integrating we obtain y 8 (y − 3) dy = dx 1 2 y − 3y = x + c 2 y 2 − 6y + 9 = 2x + c1 6 4 2 (y − 3)2 = 2x + c1 √ y = 3 ± 2x + c1 . -1 1 -2 36 2 3 4 5 x 2.2 Separable Variables The initial condition dictates whether to use the plus or minus sign. √ When y1 (0) = 4 we have c1 = 1 and y1 (x) = 3 + 2x + 1 . √ When y2 (0) = 2 we have c1 = 1 and y2 (x) = 3 − 2x + 1 . √ When y3 (1) = 2 we have c1 = −1 and y3 (x) = 3 − 2x − 1 . √ When y4 (−1) = 4 we have c1 = 3 and y4 (x) = 3 + 2x + 3 . 39. (a) Separating variables we have 2y dy = (2x + 1)dx. Integrating gives y 2 = x2 + x + c. When y(−2) = −1 we √ find c = −1, so y 2 = x2 + x − 1 and y = − x2 + x − 1 . The negative square root is chosen because of the initial condition. y 2 (b) From the figure, the largest interval of definition appears to be approximately (−∞, −1.65). 1 -5 -4 -3 -2 -1 -1 1 2 x -2 -3 -4 -5 √ ± 5 , so the largest interval of definition is (−∞, − 21 − 12 5 ). √ The right-hand endpoint of the interval is excluded because y = − x2 + x − 1 is not differentiable at this point. 2 (c) Solving x + x − 1 = 0 we get x = − 12 1 2 √ 40. (a) From Problem 7 the general solution is 3e−2y + 2e3x = c. When y(0) = 0 we find c = 5, so 3e−2y + 2e3x = 5. Solving for y we get y = − 12 ln 13 (5 − 2e3x ). y (b) The interval of definition appears to be approximately (−∞, 0.3). 2 1 x -2 -1.5 -1 -0.5 -1 -2 (c) Solving 13 (5 − 2e3x ) = 0 we get x = 13 ln( 25 ), so the exact interval of definition is (−∞, 13 ln 52 ). √ 41. (a) While y2 (x) = − 25 − x2 is defined at x = −5 and x = 5, y2′ (x) is not defined at these values, and so the interval of definition is the open interval (−5, 5). (b) At any point on the x-axis the derivative of y(x) is undefined, so no solution curve can cross the x-axis. Since −x/y is not defined when y = 0, the initial-value problem has no solution. 42. (a) Separating variables and integrating we obtain x2 − y 2 = c. For c ̸= 0 the graph is a hyperbola centered at the origin. All four initial conditions imply c = 0 and y = ±x. Since the differential equation is not defined for y = 0, solutions are y = ±x, x < 0 and y = ±x, x > 0. The solution for y(a) = a is y = x, x > 0; for y(a) = −a is y = −x; for y(−a) = a is y = −x, x < 0; and for y(−a) = −a is y = x, x < 0. (b) Since x/y is not defined when y = 0, the initial-value problem has no solution. (c) Setting x = 1 and y = 2 in x2 − y 2 = c we get c = −3, so y 2 = x2 + 3 and y(x) = √ x2 + 3 , where the positive square root is chosen because of the initial condition. The domain is all real numbers since x2 + 3 > 0 for all x. 37 2.2 Separable Variables + 1 + y 2 sin2 y = dx which is not readily integrated (even by a CAS). We note that 43. Separating variables we have dy/ *' y 3.5 3 2.5 2 1.5 1 0.5 dy/dx ≥ 0 for all values of x and y and that dy/dx = 0 when y = 0 and y = π, which are equilibrium solutions. -6 -4 -2 2 4 6 8 x . √ √ √ 44. Separating variables we have dy/( y + y) = dx/( x + x). To integrate dx/( x + x) we substitute u2 = x and get √ 2u 2 du = du = 2 ln |1 + u| + c = 2 ln(1 + x ) + c. u + u2 1+u Integrating the separated differential equation we have √ √ √ √ 2 ln(1 + y ) = 2 ln(1 + x ) + c or ln(1 + y ) = ln(1 + x ) + ln c1 . √ Solving for y we get y = [c1 (1 + x ) − 1]2 . - 45. We are looking for a function y(x) such that y2 + $ dy dx &2 = 1. Using the positive square root gives ' dy dy = dx =⇒ sin−1 y = x + c. = 1 − y 2 =⇒ ' dx 1 − y2 Thus a solution is y = sin(x + c). If we use the negative square root we obtain y = sin(c − x) = − sin(x − c) = − sin(x + c1 ). Note that when c = c1 = 0 and when c = c1 = π/2 we obtain the well known particular solutions y = sin x, y = − sin x, y = cos x, and y = − cos x. Note also that y = 1 and y = −1 are singular solutions. y 46. (a) 3 −3 3 x −3 (b) For |x| > 1 and |y| > 1 the differential equation is dy/dx = integrating, we obtain dy ' y2 −1 =√ dx x2 − 1 and ' √ y 2 − 1 / x2 − 1 . Separating variables and cosh−1 y = cosh−1 x + c. Setting x = 2 and y = 2 we find c = cosh−1 2 − cosh−1 2 = 0 and cosh−1 y = cosh−1 x. An explicit solution is y = x. 47. Since the tension T1 (or magnitude T1 ) acts at the lowest point of the cable, we use symmetry to solve the problem on the interval [0, L/2]. The assumption that the roadbed is uniform (that is, weighs a constant ρ 38 2.2 Separable Variables pounds per horizontal foot) implies W = ρx, where x is measured in feet and 0 ≤ x ≤ L/2. Therefore (10) becomes dy/dx = (ρ/T1 )x. This last equation is a separable equation of the form given in (1) of Section 2.2 in the text. Integrating and using the initial condition y(0) = a shows that the shape of the cable is a parabola: y(x) = (ρ/2T1 )x2 + a. In terms of the sag h of the cable and the span L, we see from Figure 2.22 in the text that y(L/2) = h + a. By applying this last condition to y(x) = (ρ/2T1 )x2 + a enables us to express ρ/2T1 in terms of h and L: y(x) = (4h/L2 )x2 + a. Since y(x) is an even function of x, the solution is valid on −L/2 ≤ x ≤ L/2. 48. (a) Separating variables and integrating, we have (3y 2 +1)dy = −(8x+5)dx and y 3 + y = −4x2 − 5x + c. Using a CAS we show various contours of f (x, y) = y 3 + y + 4x2 + 5x. The plots shown on [−5, 5] × [−5, 5] correspond to c-values of 0, ±5, ±20, ±40, ±80, and ±125. y 4 2 0 x -2 -4 -4 (b) The value of c corresponding to y(0) = −1 is f (0, −1) = −2; to y(0) = 2 is f (0, 2) = 10; to y(−1) = 4 is f (−1, 4) = 67; and to y(−1) = −3 is −31. -2 0 2 4 y 4 2 x 0 -2 -4 -4 -2 0 2 4 49. (a) An implicit solution of the differential equation (2y + 2)dy − (4x3 + 6x)dx = 0 is y 2 + 2y − x4 − 3x2 + c = 0. The condition y(0) = −3 implies that c = −3. Therefore y 2 + 2y − x4 − 3x2 − 3 = 0. (b) Using the quadratic formula we can solve for y in terms of x: y= −2 ± ' 4 + 4(x4 + 3x2 + 3) . 2 The explicit solution that satisfies the initial condition is then y = −1 − ' x4 + 3x3 + 4 . (c) From the graph of the function f (x) = x4 + 3x3 + 4 below we see that f (x) ≤ 0 on the approximate interval −2.8 ≤ x ≤ −1.3. Thus the approximate domain of the function y = −1 − ' ' x4 + 3x3 + 4 = −1 − f (x) is x ≤ −2.8 or x ≥ −1.3. The graph of this function is shown below. 39 2.2 Separable Variables %&&&&&&&&&&&& !1! f #x$ f#x$ -4 4 -2 x 2 -2 2 -4 -4 x -2 -6 -2 -8 -4 -10 %&&&&&&&&&&&& !1! f #x$ (d) Using the root finding capabilities of a CAS, the zeros of f are found to be −2.82202 and −1.3409. The domain of definition of the solution y(x) is then x > −1.3409. The equality has been removed since the derivative dy/dx does not exist at the points where x 2 -2 f (x) = 0. The graph of the solution y = φ(x) is given on the right. -4 -6 -8 -10 50. (a) Separating variables and integrating, we have and y (−2y + y 2 )dy = (x − x2 )dx 4 1 1 1 −y 2 + y 3 = x2 − x3 + c. 3 2 3 Using a CAS we show some contours of 2 -2 f (x, y) = 2y 3 − 6y 2 + 2x3 − 3x2 . The plots shown on [−7, 7]×[−5, 5] correspond to c-values -4 of −450, −300, −200, −120, −60, −20, −10, −8.1, −5, −0.8, 20, 60, and 120. + * (b) The value of c corresponding to y(0) = 23 is f 0, 32 = − 27 4 . The portion of the graph between the dots corresponds to the solution curve satisfying the intial condition. To determine the interval of definition we find dy/dx for 2y 3 − 6y 2 + 2x3 − 3x2 = − x 0 -6 -2 0 2 4 6 y 4 2 0 27 . 4 -4 x -2 Using implicit differentiation we get y ′ = (x − x2 )/(y 2 − 2y), -4 which is infinite when y = 0 and y = 2. Letting y = 0 in -2 0 2 4 6 2y 3 − 6y 2 + 2x3 − 3x2 = − 27 and using a CAS to solve for x 4 we get x = −1.13232. Similarly, letting y = 2, we find x = 1.71299. The largest interval of definition is approximately (−1.13232, 1.71299). 40 2.3 (c) The value of c corresponding to y(0) = −2 is f (0, −2) = −40. The portion of the graph to the right of the dot corresponds to the solution curve satisfying the initial condition. To determine the interval of definition we find dy/dx for y 4 2 x 0 -2 2y 3 − 6y 2 + 2x3 − 3x2 = −40. ′ Linear Equations -4 2 2 Using implicit differentiation we get y = (x − x )/(y − 2y), which is infinite when y = 0 and y = 2. Letting y = 0 in -6 -8 -4 -2 0 2 4 2y 3 − 6y 2 + 2x3 − 3x2 = −40 and using a CAS to solve for x we get x = −2.29551. The largest interval of definition is approximately (−2.29551, ∞). 6 8 10 EXERCISES 2.3 Linear Equations . d / −5x 0 e y = 0 and y = ce5x for −∞ < x < ∞. 1. For y ′ − 5y = 0 an integrating factor is e− 5 dx = e−5x so that dx . d / 2x 0 2. For y ′ + 2y = 0 an integrating factor is e 2 dx = e2x so that e y = 0 and y = ce−2x for −∞ < x < ∞. dx The transient term is ce−2x . . d x [e y] = e4x and y = 14 e3x +ce−x for −∞ < x < ∞. 3. For y ′ +y = e3x an integrating factor is e dx = ex so that dx The transient term is ce−x . . d / 4x 0 4. For y ′ + 4y = 34 an integrating factor is e 4 dx = e4x so that e y = 43 e4x and y = 31 + ce−4x for dx −∞ < x < ∞. The transient term is ce−4x . . 2 3 d 1 x3 2 3 3 e y = x2 ex and y = 13 + ce−x for 5. For y ′ + 3x2 y = x2 an integrating factor is e 3x dx = ex so that dx 3 −∞ < x < ∞. The transient term is ce−x . . 2 d 1 x2 2 2 2 6. For y ′ + 2xy = x3 an integrating factor is e 2x dx = ex so that e y = x3 ex and y = 12 x2 − 21 + ce−x dx 2 for −∞ < x < ∞. The transient term is ce−x . . 1 d 1 1 1 c 7. For y ′ + y = 2 an integrating factor is e (1/x)dx = x so that [xy] = and y = ln x + for 0 < x < ∞. x x dx x x x . / 0 d 8. For y ′ − 2y = x2 + 5 an integrating factor is e− 2 dx = e−2x so that e−2x y = x2 e−2x + 5e−2x and dx 2x y = − 21 x2 − 21 x − 11 for −∞ < x < ∞. 4 + ce ( ) . 1 d 1 1 − (1/x)dx ′ = so that y = sin x and y = cx − x cos x for 9. For y − y = x sin x an integrating factor is e x x dx x 0 < x < ∞. . d / 2 0 3 2 ′ 10. For y + y = an integrating factor is e (2/x)dx = x2 so that x y = 3x and y = 23 +cx−2 for 0 < x < ∞. x x dx . d / 4 0 4 x y = x6 −x4 and y = 17 x3 − 51 x+cx−4 11. For y ′ + y = x2 −1 an integrating factor is e (4/x)dx = x4 so that x dx for 0 < x < ∞. 41 2.3 Linear Equations . 0 d / x y = x an integrating factor is e− [x/(1+x)]dx = (x+1)e−x so that (x + 1)e−x y = x(x+1)e−x (1 + x) dx 2x + 3 cex and y = −x − + for −1 < x < ∞. x+1 x+1 & $ . d 2 x ex 2 y = 2 an integrating factor is e [1+(2/x)]dx = x2 ex so that [x e y] = e2x and 13. For y ′ + 1 + x x dx 1 ex ce−x ce−x y= + for 0 < x < ∞. The transient term is . 2 x2 x2 x2 $ & . 1 1 d ′ 14. For y + 1 + y = e−x sin 2x an integrating factor is e [1+(1/x)]dx = xex so that [xex y] = sin 2x and x x dx ce−x 1 y = − e−x cos 2x + for 0 < x < ∞. The entire solution is transient. 2x x . d / −4 0 dx 4 −4 15. For − x = 4y 5 an integrating factor is e− (4/y)dy = eln y = y −4 so that y x = 4y and x = 2y 6 +cy 4 dy y dy 12. For y ′ − for 0 < y < ∞. . d / 2 0 2 2 c dx 2 + x = ey an integrating factor is e (2/y)dy = y 2 so that y x = y 2 ey and x = ey − ey + 2 ey + 2 dy y dy y y y c for 0 < y < ∞. The transient term is 2 . y . d 17. For y ′ + (tan x)y = sec x an integrating factor is e tan x dx = sec x so that [(sec x)y] = sec2 x and dx y = sin x + c cos x for −π/2 < x < π/2. . d 18. For y ′ +(cot x)y = sec2 x csc x an integrating factor is e cot x dx = eln | sin x| = sin x so that [(sin x) y] = sec2 x dx and y = sec x + c csc x for 0 < x < π/2. . x+2 d 2xe−x 19. For y ′ + y= an integrating factor is e [(x+2)/(x+1)]dx = (x + 1)ex , so [(x + 1)ex y] = 2x and x+1 x+1 dx 16. For c x2 −x e + e−x for −1 < x < ∞. The entire solution is transient. x+1 x+1 . 0 4 d / 5 [4/(x+2)]dx For y ′ + = (x + 2)4 so that y= an integrating factor is e (x + 2)4 y = 5(x + 2)2 2 x+2 (x + 2) dx 5 −1 −4 and y = (x + 2) + c(x + 2) for −2 < x < ∞. The entire solution is transient. 3 . dr For + r sec θ = cos θ an integrating factor is e sec θ dθ = eln | sec x+tan x| = sec θ + tan θ so that dθ d [(sec θ + tan θ)r] = 1 + sin θ and (sec θ + tan θ)r = θ − cos θ + c for −π/2 < θ < π/2 . dθ . 2 dP d 1 t2 −t 2 2 For + (2t − 1)P = 4t − 2 an integrating factor is e (2t−1) dt = et −t so that e P = (4t − 2)et −t and dt dt 2 2 P = 2 + cet−t for −∞ < t < ∞. The transient term is cet−t . $ & . 1 d / 3x 0 e−3x For y ′ + 3 + y= an integrating factor is e [3+(1/x)]dx = xe3x so that xe y = 1 and x x dx ce−3x y = e−3x + for 0 < x < ∞. The transient term is ce−3x /x. x ( ) . 2 x−1 x+1 d x−1 [2/(x2 −1)]dx ′ For y + 2 = y = an integrating factor is e so that y = 1 and x −1 x−1 x+1 dx x + 1 (x − 1)y = x(x + 1) + c(x + 1) for −1 < x < 1. y= 20. 21. 22. 23. 24. 42 2.3 Linear Equations . 1 d 1 c 1 y = ex an integrating factor is e (1/x)dx = x so that [xy] = ex and y = ex + for 0 < x < ∞. x x dx x x 1 2−e If y(1) = 2 then c = 2 − e and y = ex + . x x ( ) . 1 dx 1 d 1 − (1/y)dy − x = 2y an integrating factor is e = so that x = 2 and x = 2y 2 +cy for 0 < y < ∞. For dy y y dy y 49 If y(1) = 5 then c = −49/5 and x = 2y 2 − y. 5 . di R d 1 Rt/L 2 E Rt/L E For i = e and + i= an integrating factor is e (R/L) dt = eRt/L so that e dt L L dt L & $ E E E i= e−Rt/L . + ce−Rt/L for −∞ < t < ∞. If i(0) = i0 then c = i0 − E/R and i = + i0 − R R R . d −kt dT −kT = −Tm k an integrating factor is e (−k)dt = e−kt so that [e T ] = −Tm ke−kt and T = Tm +cekt For dt dt for −∞ < t < ∞. If T (0) = T0 then c = T0 − Tm and T = Tm + (T0 − Tm )ekt . . 1 d ln x For y ′ + y = an integrating factor is e [1/(x+1)]dx = x + 1 so that [(x + 1)y] = ln x and x+1 x+1 dx x c x x 21 x ln x − + for 0 < x < ∞. If y(1) = 10 then c = 21 and y = ln x − + . y= x+1 x+1 x+1 x+1 x+1 x+1 . d For y ′ + (tan x)y = cos2 x an integrating factor is e tan x dx = eln | sec x| = sec x so that [(sec x) y] = cos x dx and y = sin x cos x + c cos x for −π/2 < x < π/2. If y(0) = −1 then c = −1 and y = sin x cos x − cos x. 25. For y ′ + 26. 27. 28. 29. 30. 31. For y ′ + 2y = f (x) an integrating factor is e2x so that ! 1 2x e + c1 , 0 ≤ x ≤ 3 ye2x = 2 c2 , x > 3. 1 y If y(0) = 0 then c1 = −1/2 and for continuity we must have c2 = 12 e6 − 21 so that 31 −2x ), 0≤x≤3 2 (1 − e y= 1 6 −2x , x > 3. 2 (e − 1)e 32. For y ′ + y = f (x) an integrating factor is ex so that ! x e + c1 , 0≤x≤1 x ye = x −e + c2 , x > 1. 1 If y(0) = 1 then c1 = 0 and for continuity we must have c2 = 2e so that ! 1, 0≤x≤1 y= 1−x 2e − 1, x > 1. x 5 y 5 x 3 x -1 2 33. For y ′ + 2xy = f (x) an integrating factor is ex so that 3 1 x2 e + c1 , 0 ≤ x ≤ 1 x2 ye = 2 c2 , x > 1. If y(0) = 2 then c1 = 3/2 and for continuity we must have c2 = 12 e + so that y= 31 2 3 −x , 0≤x≤1 2 + 2e + *1 2 3 −x , x > 1. 2e + 2 e 43 2 y 3 2 2.3 Linear Equations 34. For y′ + ⎧ x ⎪ ⎨ 1 + x2 , 0 ≤ x ≤ 1 2x y= ⎪ 1 + x2 ⎩ −x , 1 + x2 1 y x > 1, 5 x -1 an integrating factor is 1 + x2 so that 31 2 0≤x≤1 * + 2 x + c1 , 2 1+x y = − 12 x2 + c2 , x > 1. If y(0) = 0 then c1 = 0 and for continuity we must have c2 = 1 so that ⎧1 1 ⎪ ⎪ ⎨ 2 − 2 (1 + x2 ) , 0 ≤ x ≤ 1 y= ⎪ 3 1 ⎪ ⎩ − , x > 1. 2 (1 + x2 ) 2 35. We first solve the initial-value problem y ′ + 2y = 4x, y(0) = 3 on the interval [0, 1]. . 2 dx The integrating factor is e = e2x , so d 2x [e y] = 4xe2x dx 2x e y = 4xe2x dx = 2xe2x − e2x + c1 y 20 15 10 y = 2x − 1 + c1 e−2x . 5 −2x Using the initial condition, we find y(0) = −1+c1 = 3, so c1 = 4 and y = 2x−1+4e , −2 −2 0 ≤ x ≤ 1. Now, since y(1) = 2−1+4e = 1+4e , we solve the initial-value problem ′ −2 y. − (2/x)y = 4x, y(1) = 1 + 4e e (−2/x)dx = e−2 ln x = x−2 , so 3 x on the interval (1, ∞). The integrating factor is d −2 4 [x y] = 4xx−2 = dx x 4 −2 x y= dx = 4 ln x + c2 x y = 4x2 ln x + c2 x2 . (We use ln x instead of ln |x| because x > 1.) Using the initial condition we find y(1) = c2 = 1 + 4e−2 , so y = 4x2 ln x + (1 + 4e−2 )x2 , x > 1. Thus, the solution of the original initial-value problem is y= ! 2x − 1 + 4e−2x , 0≤x≤1 4x2 ln x + (1 + 4e−2 )x2 , x > 1. See Problem 42 in this section. x 36. For y ′ + ex y = 1 an integrating factor is ee . Thus x x d / ex 0 e y = ee and ee y = dx x .x t x From y(0) = 1 we get c = e, so y = e−e 0 ee dt + e1−e . - x t ee dt + c. 0 When y ′ + ex y = 0 we can separate variables and integrate: dy = −ex dx y and 44 ln |y| = −ex + c. 2.3 x Linear Equations x Thus y = c1 e−e . From y(0) = 1 we get c1 = e, so y = e1−e . When y ′ + ex y = ex we can see by inspection that y = 1 is a solution. 2 37. An integrating factor for y ′ − 2xy = 1 is e−x . Thus 2 d −x2 [e y] = e−x dx √ - x 2 2 π erf(x) + c e−t dt = e−x y = 2 0 √ 2 π x2 y= e erf(x) + cex . 2 √ √ From y(1) = ( π/2)e erf(1) + ce = 1 we get c = e−1 − 2π erf(1). The solution of the initial-value problem is $ & √ √ 2 π x2 π −1 e erf(x) + e − erf(1) ex y= 2 2 √ π x2 x2 −1 =e + e (erf(x) − erf(1)). 2 38. We want 4 to be a critical point, so we use y ′ = 4 − y. 39. (a) All solutions of the form y = x5 ex − x4 ex + cx4 satisfy the initial condition. In this case, since 4/x is discontinuous at x = 0, the hypotheses of Theorem 1.1 are not satisfied and the initial-value problem does not have a unique solution. (b) The differential equation has no solution satisfying y(0) = y0 , y0 > 0. (c) In this case, since x0 > 0, Theorem 1.1 applies and the initial-value problem has a unique solution given by y = x5 ex − x4 ex + cx4 where c = y0 /x40 − x0 ex0 + ex0 . 40. On the interval (−3, 3) the integrating factor is . . 2 e x dx/(x −9) = e− and so x dx/(9−x2 ) 1 2 = e 2 ln(9−x ) = ' 9 − x2 2 d 1' c . 9 − x2 y = 0 and y = √ dx 9 − x2 41. We want the general solution to be y = 3x − 5 + ce−x . (Rather than e−x , any function that approaches 0 as x → ∞ could be used.) Differentiating we get y ′ = 3 − ce−x = 3 − (y − 3x + 5) = −y + 3x − 2, so the differential equation y ′ + y = 3x − 2 has solutions asymptotic to the line y = 3x − 5. 42. The left-hand derivative of the function at x = 1 is 1/e and the right-hand derivative at x = 1 is 1 − 1/e. Thus, y is not differentiable at x = 1. 43. (a) Differentiating yc = c/x3 we get 3c 3 c 3 =− = − yc 4 3 x x x x so a differential equation with general solution yc = c/x3 is xy ′ + 3y = 0. Now yc′ = − xyp′ + 3yp = x(3x2 ) + 3(x3 ) = 6x3 so a differential equation with general solution y = c/x3 + x3 is xy ′ + 3y = 6x3 . This will be a general solution on (0, ∞). 45 2.3 Linear Equations (b) Since y(1) = 13 − 1/13 = 0, an initial condition is y(1) = 0. Since y(1) = 13 + 2/13 = 3, an initial condition is y(1) = 3. In each case the y 3 interval of definition is (0, ∞). The initial-value problem xy ′ + 3y = 6x3 , y(0) = 0 has solution y = x3 for −∞ < x < ∞. In the figure the lower curve is the graph of y(x) = x3 − 1/x3 ,while the upper curve is the graph 5 of y = x3 − 2/x3 . x -3 (c) The first two initial-value problems in part (b) are not unique. For example, setting y(2) = 23 − 1/23 = 63/8, we see that y(2) = 63/8 is also an initial condition leading to the solution y = x3 − 1/x3 . . . . 44. Since e P (x)dx+c = ec e P (x)dx = c1 e P (x)dx , we would have - . . . . P (x)dx P (x)dx P (x)dx c1 e y = c2 + c1 e f (x) dx and e y = c3 + e P (x)dx f (x) dx, which is the same as (6) in the text. 45. We see by inspection that y = 0 is a solution. 46. The solution of the first equation is x = c1 e−λ1 t . From x(0) = x0 we obtain c1 = x0 and so x = x0 e−λ1 t . The second equation then becomes dy = x0 λ1 e−λ1 t − λ2 y dt or dy + λ2 y = x0 λ1 e−λ1 t dt which is linear. An integrating factor is eλ2 t . Thus d λ2 t [e y ] = x0 λ1 e−λ1 t eλ2 t = x0 λ1 e(λ2 −λ1 )t dt x0 λ1 (λ2 −λ1 )t e + c2 eλ2 t y = λ2 − λ1 x0 λ1 −λ1 t y= e + c2 e−λ2 t . λ2 − λ1 From y(0) = y0 we obtain c2 = (y0 λ2 − y0 λ1 − x0 λ1 )/(λ2 − λ1 ). The solution is y= 47. Writing the differential equation as x0 λ1 −λ1 t y0 λ2 − y0 λ1 − x0 λ1 −λ2 t e + e . λ2 − λ 1 λ2 − λ1 dE 1 + E = 0 we see that an integrating factor is et/RC . Then dt RC d t/RC [e E] = 0 dt et/RC E = c E = ce−t/RC . From E(4) = ce−4/RC = E0 we find c = E0 e4/RC . Thus, the solution of the initial-value problem is E = E0 e4/RC e−t/RC = E0 e−(t−4)/RC . 46 2.3 Linear Equations 2 48. (a) An integrating factor for y ′ − 2xy = −1 is e−x . Thus 2 d −x2 [e y] = −e−x dx √ - x π −t2 −x2 erf(x) + c. e dt = − e y=− 2 0 √ √ From y(0) = π/2, and noting that erf(0) = 0, we get c = π/2. Thus $ √ √ & √ √ 2 π π π x2 π x2 erf(x) + e (1 − erf(x)) = e erfc(x). y = ex − = 2 2 2 2 y (b) Using a CAS we find y(2) ≈ 0.226339. 5 5 x 49. (a) An integrating factor for y′ + is x2 . Thus 2 10 sin x y= x x3 d 2 sin x [x y] = 10 dx x - x sin t 2 dt + c x y = 10 t 0 y = 10x−2 Si(x) + cx−2 . From y(1) = 0 we get c = −10Si(1). Thus y = 10x−2 Si(x) − 10x−2 Si(1) = 10x−2 (Si(x) − Si(1)). (b) y 2 1 -1 -2 -3 -4 -5 1 2 3 4 5 x (c) From the graph in part (b) we see that the absolute maximum occurs around x = 1.7. Using the root-finding capability of a CAS and solving y ′ (x) = 0 for x we see that the absolute maximum is (1.688, 1.742). .x − sin t2 dt 50. (a) The integrating factor for y ′ − (sin x2 )y = 0 is e 0 . Then .x d − sin t2 dt y] = 0 [e 0 dx .x − sin t2 dt e 0 y = c1 .x sin t2 dt y = c1 e 0 . 47 2.3 Linear Equations ' ' π/2 du and % % - √2/π x $% & - x "π # π π 2 2 2 u du = S x sin sin t dt = 2 0 2 2 π 0 √ √ √ √ so y = c1 e π/2 S( 2/π x) . Using S(0) = 0 and y(0) = c1 = 5 we have y = 5e π/2 S( 2/π x) . Letting t = π/2 u we have dt = (b) y 10 5 -10 -5 x 5 10 (c) From the graph we see that as x → ∞, y(x) oscillates with decreasing amplitudes approaching 9.35672. √ Since limx→∞ 5S(x) = 21 , we have limx→∞ y(x) = 5e π/8 ≈ 9.357, and since limx→−∞ S(x) = − 12 , we √ have limx→−∞ y(x) = 5e− π/8 ≈ 2.672. (d) From the graph in part (b) we see that the absolute maximum occurs around x = 1.7 and the absolute minimum occurs around x = −1.8. Using the root-finding capability of a CAS and solving y ′ (x) = 0 for x, we see that the absolute maximum is (1.772, 12.235) and the absolute minimum is (−1.772, 2.044). EXERCISES 2.4 Exact Equations 1. Let M = 2x − 1 and N = 3y + 7 so that My = 0 = Nx . From fx = 2x − 1 we obtain f = x2 − x + h(y), h′ (y) = 3y + 7, and h(y) = 32 y 2 + 7y. A solution is x2 − x + 32 y 2 + 7y = c. 2. Let M = 2x + y and N = −x − 6y. Then My = 1 and Nx = −1, so the equation is not exact. 3. Let M = 5x + 4y and N = 4x − 8y 3 so that My = 4 = Nx . From fx = 5x + 4y we obtain f = 25 x2 + 4xy + h(y), h′ (y) = −8y 3 , and h(y) = −2y 4 . A solution is 52 x2 + 4xy − 2y 4 = c. 4. Let M = sin y − y sin x and N = cos x + x cos y − y so that My = cos y − sin x = Nx . From fx = sin y − y sin x we obtain f = x sin y + y cos x + h(y), h′ (y) = −y, and h(y) = − 12 y 2 . A solution is x sin y + y cos x − 21 y 2 = c. 5. Let M = 2y 2 x−3 and N = 2yx2 +4 so that My = 4xy = Nx . From fx = 2y 2 x−3 we obtain f = x2 y 2 −3x+h(y), h′ (y) = 4, and h(y) = 4y. A solution is x2 y 2 − 3x + 4y = c. 6. Let M = 4x3 −3y sin 3x−y/x2 and N = 2y−1/x+cos 3x so that My = −3 sin 3x−1/x2 and Nx = 1/x2 −3 sin 3x. The equation is not exact. 7. Let M = x2 − y 2 and N = x2 − 2xy so that My = −2y and Nx = 2x − 2y. The equation is not exact. 8. Let M = 1 + ln x + y/x and N = −1 + ln x so that My = 1/x = Nx . From fy = −1 + ln x we obtain f = −y + y ln x + h(y), h′ (x) = 1 + ln x, and h(y) = x ln x. A solution is −y + y ln x + x ln x = c. 48 2.4 Exact Equations 9. Let M = y 3 − y 2 sin x − x and N = 3xy 2 + 2y cos x so that My = 3y 2 − 2y sin x = Nx . From fx = y 3 − y 2 sin x − x we obtain f = xy 3 + y 2 cos x − 21 x2 + h(y), h′ (y) = 0, and h(y) = 0. A solution is xy 3 + y 2 cos x − 12 x2 = c. 10. Let M = x3 + y 3 and N = 3xy 2 so that My = 3y 2 = Nx . From fx = x3 + y 3 we obtain f = 14 x4 + xy 3 + h(y), h′ (y) = 0, and h(y) = 0. A solution is 14 x4 + xy 3 = c. 11. Let M = y ln y − e−xy and N = 1/y + x ln y so that My = 1 + ln y + xe−xy and Nx = ln y. The equation is not exact. 12. Let M = 3x2 y + ey and N = x3 + xey − 2y so that My = 3x2 + ey = Nx . From fx = 3x2 y + ey we obtain f = x3 y + xey + h(y), h′ (y) = −2y, and h(y) = −y 2 . A solution is x3 y + xey − y 2 = c. 13. Let M = y − 6x2 − 2xex and N = x so that My = 1 = Nx . From fx = y − 6x2 − 2xex we obtain f = xy − 2x3 − 2xex + 2ex + h(y), h′ (y) = 0, and h(y) = 0. A solution is xy − 2x3 − 2xex + 2ex = c. 14. Let M = 1 − 3/x + y and N = 1 − 3/y + x so that My = 1 = Nx . From fx = 1 − 3/x + y we obtain 3 f = x − 3 ln |x| + xy + h(y), h′ (y) = 1 − , and h(y) = y − 3 ln |y|. A solution is x + y + xy − 3 ln |xy| = c. y + + * * 15. Let M = x2 y 3 − 1/ 1 + 9x2 and N = x3 y 2 so that My = 3x2 y 2 = Nx . From fx = x2 y 3 − 1/ 1 + 9x2 we obtain f = 13 x3 y 3 − 13 arctan(3x) + h(y), h′ (y) = 0, and h(y) = 0. A solution is x3 y 3 − arctan(3x) = c. 16. Let M = −2y and N = 5y − 2x so that My = −2 = Nx . From fx = −2y we obtain f = −2xy + h(y), h′ (y) = 5y, and h(y) = 25 y 2 . A solution is −2xy + 25 y 2 = c. 17. Let M = tan x − sin x sin y and N = cos x cos y so that My = − sin x cos y = Nx . From fx = tan x − sin x sin y we obtain f = ln | sec x| + cos x sin y + h(y), h′ (y) = 0, and h(y) = 0. A solution is ln | sec x| + cos x sin y = c. 2 2 18. Let M = 2y sin x cos x − y + 2y 2 exy and N = −x + sin2 x + 4xyexy so that 2 2 My = 2 sin x cos x − 1 + 4xy 3 exy + 4yexy = Nx . 2 2 From fx = 2y sin x cos x − y + 2y 2 exy we obtain f = y sin2 x − xy + 2exy + h(y), h′ (y) = 0, and h(y) = 0. A 2 solution is y sin2 x − xy + 2exy = c. 19. Let M = 4t3 y − 15t2 − y and N = t4 + 3y 2 − t so that My = 4t3 − 1 = Nt . From ft = 4t3 y − 15t2 − y we obtain f = t4 y − 5t3 − ty + h(y), h′ (y) = 3y 2 , and h(y) = y 3 . A solution is t4 y − 5t3 − ty + y 3 = c. * + * + * + * +2 20. Let M = 1/t + 1/t2 − y/ t2 + y 2 and N = yey + t/ t2 + y 2 so that My = y 2 − t2 / t2 + y 2 = Nt . From $ & + * 1 t + h(y), h′ (y) = yey , and h(y) = yey − ey . ft = 1/t + 1/t2 − y/ t2 + y 2 we obtain f = ln |t| − − arctan t y A solution is $ & 1 t ln |t| − − arctan + yey − ey = c. t y 21. Let M = x2 + 2xy + y 2 and N = 2xy + x2 − 1 so that My = 2(x + y) = Nx . From fx = x2 + 2xy + y 2 we obtain f = 31 x3 + x2 y + xy 2 + h(y), h′ (y) = −1, and h(y) = −y. The solution is 31 x3 + x2 y + xy 2 − y = c. If y(1) = 1 then c = 4/3 and a solution of the initial-value problem is 1 3 3x + x2 y + xy 2 − y = 4 3 . 22. Let M = ex + y and N = 2 + x + yey so that My = 1 = Nx . From fx = ex + y we obtain f = ex + xy + h(y), h′ (y) = 2 + yey , and h(y) = 2y + yey − y. The solution is ex + xy + 2y + yey − ey = c. If y(0) = 1 then c = 3 and a solution of the initial-value problem is ex + xy + 2y + yey − ey = 3. 23. Let M = 4y + 2t − 5 and N = 6y + 4t − 1 so that My = 4 = Nt . From ft = 4y + 2t − 5 we obtain f = 4ty + t2 − 5t + h(y), h′ (y) = 6y − 1, and h(y) = 3y 2 − y. The solution is 4ty + t2 − 5t + 3y 2 − y = c. If y(−1) = 2 then c = 8 and a solution of the initial-value problem is 4ty + t2 − 5t + 3y 2 − y = 8. 49 2.4 Exact Equations + * t2 24. Let M = t/2y 4 and N = 3y 2 − t2 /y 5 so that My = −2t/y 5 = Nt . From ft = t/2y 4 we obtain f = 4 + h(y), 4y 2 3 3 t 3 h′ (y) = 3 , and h(y) = − 2 . The solution is 4 − 2 = c. If y(1) = 1 then c = −5/4 and a solution of the y 2y 4y 2y t2 3 5 − 2 =− . initial-value problem is 4y 4 2y 4 25. Let M = y 2 cos x − 3x2 y − 2x and N = 2y sin x − x3 + ln y so that My = 2y cos x − 3x2 = Nx . From fx = y 2 cos x − 3x2 y − 2x we obtain f = y 2 sin x − x3 y − x2 + h(y), h′ (y) = ln y, and h(y) = y ln y − y. The solution is y 2 sin x − x3 y − x2 + y ln y − y = c. If y(0) = e then c = 0 and a solution of the initial-value problem is y 2 sin x − x3 y − x2 + y ln y − y = 0. * + 26. Let M = y 2 + y sin x and N = 2xy − cos x − 1/ 1 + y 2 so that My = 2y + sin x = Nx . From fx = y 2 + y sin x we −1 , and h(y) = − tan−1 y. The solution is xy 2 −y cos x−tan−1 y = c. obtain f = xy 2 −y cos x+h(y), h′ (y) = 1 + y2 π If y(0) = 1 then c = −1 − π/4 and a solution of the initial-value problem is xy 2 − y cos x − tan−1 y = −1 − . 4 27. Equating My = 3y 2 + 4kxy 3 and Nx = 3y 2 + 40xy 3 we obtain k = 10. 28. Equating My = 18xy 2 − sin y and Nx = 4kxy 2 − sin y we obtain k = 9/2. 29. Let M = −x2 y 2 sin x + 2xy 2 cos x and N = 2x2 y cos x so that My = −2x2 y sin x + 4xy cos x = Nx . From fy = 2x2 y cos x we obtain equation is x2 y 2 cos x = c. f = x2 y 2 cos x + h(y), h′ (y) = 0, and h(y) = 0. A solution of the differential 30. Let M = (x2 +2xy−y 2 )/(x2 +2xy+y 2 ) and N = (y 2 +2xy−x2 /(y 2 +2xy+x2 ) so that My = −4xy/(x+y)3 = Nx . + * 2y 2 From fx = x2 + 2xy + y 2 − 2y 2 /(x + y)2 we obtain f = x + + h(y), h′ (y) = −1, and h(y) = −y. A x+y solution of the differential equation is x2 + y 2 = c(x + y). . 31. We note that (My − Nx )/N = 1/x, so an integrating factor is e dx/x = x. Let M = 2xy 2 + 3x2 and N = 2x2 y so that My = 4xy = Nx . From fx = 2xy 2 + 3x2 we obtain f = x2 y 2 + x3 + h(y), h′ (y) = 0, and h(y) = 0. A solution of the differential equation is x2 y 2 + x3 = c. . 32. We note that (My − Nx )/N = 1, so an integrating factor is e dx = ex . Let M = xyex + y 2 ex + yex and N = xex + 2yex so that My = xex + 2yex + ex = Nx . From fy = xex + 2yex we obtain f = xyex + y 2 ex + h(x), h′ (y) = 0, and h(y) = 0. A solution of the differential equation is xyex + y 2 ex = c. . 33. We note that (Nx −My )/M = 2/y, so an integrating factor is e 2dy/y = y 2 . Let M = 6xy 3 and N = 4y 3 +9x2 y 2 so that My = 18xy 2 = Nx . From fx = 6xy 3 we obtain f = 3x2 y 3 + h(y), h′ (y) = 4y 3 , and h(y) = y 4 . A solution of the differential equation is 3x2 y 3 + y 4 = c. . 34. We note that (My −Nx )/N = − cot x, so an integrating factor is e− cot x dx = csc x. Let M = cos x csc x = cot x and N = (1 + 2/y) sin x csc x = 1 + 2/y, so that My = 0 = Nx . From fx = cot x we obtain f = ln(sin x) + h(y), h′ (y) = 1 + 2/y, and h(y) = y + ln y 2 . A solution of the differential equation is ln(sin x) + y + ln y 2 = c. . 35. We note that (My − Nx )/N = 3, so an integrating factor is e 3 dx = e3x . Let M = (10 − 6y + e−3x )e3x = 10e3x − 6ye3x + 1 and N = −2e3x , so that My = −6e3x = Nx . From fx = 10e3x − 6ye3x + 1 we obtain f = 10 3x 3x 3 e − 2ye + x + h(y), 3x 3x h′ (y) = 0, and h(y) = 0. A solution of the differential equation is 10 3 e − 2ye + x = c. . 36. We note that (Nx − My )/M = −3/y, so an integrating factor is e−3 dy/y = 1/y 3 . Let M = (y 2 + xy 3 )/y 3 = 1/y + x and N = (5y 2 − xy + y 3 sin y)/y 3 = 5/y − x/y 2 + sin y, so that My = −1/y 2 = Nx . From fx = 1/y + x we obtain f = x/y + 12 x2 + h(y), h′ (y) = 5/y + sin y, and h(y) = 5 ln |y| − cos y. A solution of the differential equation is x/y + 21 x2 + 5 ln |y| − cos y = c. 50 2.4 Exact Equations . 2 37. We note that (My − Nx )/N = 2x/(4 + x2 ), so an integrating factor is e−2 x dx/(4+x ) = 1/(4 + x2 ). Let M = x/(4 + x2 ) and N = (x2 y + 4y)/(4 + x2 ) = y, so that My = 0 = Nx . From fx = x(4 + x2 ) we obtain f = 21 ln(4+x2 )+h(y), h′ (y) = y, and h(y) = 12 y 2 . A solution of the differential equation is 21 ln(4+x2 )+ 12 y 2 = c. . 38. We note that (My − Nx )/N = −3/(1 + x), so an integrating factor is e−3 dx/(1+x) = 1/(1 + x)3 . Let M = (x2 + y 2 − 5)/(1 + x)3 and N = −(y + xy)/(1 + x)3 = −y/(1 + x)2 , so that My = 2y/(1 + x)3 = Nx . From fy = −y/(1 + x)2 we obtain f = − 21 y 2 /(1 + x)2 + h(x), h′ (x) = (x2 − 5)/(1 + x)3 , and h(x) = 2/(1 + x)2 + 2/(1 + x) + ln |1 + x|. A solution of the differential equation is − 2 2 y2 + + + ln |1 + x| = c. 2(1 + x)2 (1 + x)2 (1 + x) 39. (a) Implicitly differentiating x3 + 2x2 y + y 2 = c and solving for dy/dx we obtain 3x2 + 2x2 dy dy + 4xy + 2y = 0 and dx dx dy 3x2 + 4xy =− 2 . dx 2x + 2y By writing the last equation in differential form we get (4xy + 3x2 )dx + (2y + 2x2 )dy = 0. (b) Setting x = 0 and y = −2 in x3 + 2x2 y + y 2 = c we find c = 4, and setting x = y = 1 we also find c = 4. Thus, both initial conditions determine the same implicit solution. y 4 (c) Solving x3 + 2x2 y + y 2 = 4 for y we get ' y1 (x) = −x2 − 4 − x3 + x4 2 y2 and y2 (x) = −x2 + ' -4 4 − x3 + x4 . Observe in the figure that y1 (0) = −2 and y2 (1) = 1. -2 2 -2 4 x y1 -4 -6 40. To see that the equations are not equivalent consider dx = −(x/y)dy. An integrating factor is µ(x, y) = y resulting in y dx + x dy = 0. A solution of the latter equation is y = 0, but this is not a solution of the original equation. ' (3 + cos2 x)/(1 − x2 ) . Since 3 + cos2 x > 0 for all x we must have 1 − x2 > 0 or −1 < x < 1. Thus, the interval of definition is (−1, 1). y y 42. (a) Since fy = N (x, y) = xexy +2xy+1/x we obtain f = exy +xy 2 + +h(x) so that fx = yexy +y 2 − 2 +h′ (x). x x y Let M (x, y) = yexy + y 2 − 2 . x , * +−1 1 , (b) Since fx = M (x, y) = y 1/2 x−1/2 + x x2 + y we obtain f = 2y 1/2 x1/2 + ln ,x2 + y , + g(y) so that 2 +−1 +−1 1* 2 1* 2 + g ′ (x). Let N (x, y) = y −1/2 x1/2 + . x +y x +y fy = y −1/2 x1/2 + 2 2 43. First note that "' # y x d dx + ' dy. x2 + y 2 = ' 2 2 2 x +y x + y2 ' Then x dx + y dy = x2 + y 2 dx becomes # "' x y ' x2 + y 2 = dx. dx + ' dy = d x2 + y 2 x2 + y 2 41. The explicit solution is y = 51 2.4 Exact Equations ' The left side is the total differential of x2 + y 2 and the right side is the total differential of x + c. Thus ' x2 + y 2 = x + c is a solution of the differential equation. 44. To see that the statement is true, write the separable equation as −g(x) dx+dy/h(y) = 0. Identifying M = −g(x) and N = 1/h(y), we see that My = 0 = Nx , so the differential equation is exact. 45. (a) In differential form we have (v 2 − 32x)dx + xv dv = 0. This is not an exact form, but µ(x) = x is an integrating factor. Multiplying by x we get (xv 2 − 32x2 )dx + x2 v dv = 0. This form is the total differential 1 2 2 32 3 3 of u = 21 x2 v 2 − 32 3 x , so an implicit solution is 2 x v − 3 x = c. Letting x = 3 and v = 0 we find c = −288. Solving for v we get % v=8 x 9 − 2. 3 x (b) The chain leaves the platform when x = 8, so the velocity at this time is % 8 9 v(8) = 8 − ≈ 12.7 ft/s. 3 64 46. (a) Letting M (x, y) = (x2 2xy + y 2 )2 and N (x, y) = 1 + y 2 − x2 (x2 + y 2 )2 we compute My = 2x3 − 8xy 2 = Nx , (x2 + y 2 )3 so the differential equation is exact. Then we have ∂f 2xy = M (x, y) = 2 = 2xy(x2 + y 2 )−2 ∂x (x + y 2 )2 y f (x, y) = −y(x2 + y 2 )−1 + g(y) = − 2 + g(y) x + y2 ∂f y 2 − x2 y 2 − x2 + g ′ (y) = N (x, y) = 1 + 2 . = 2 2 2 ∂y (x + y ) (x + y 2 )2 y = c. When c = 0 the solution is x2 + y 2 = 1. Thus, g ′ (y) = 1 and g(y) = y. The solution is y − 2 x + y2 (b) The first graph below is obtained in Mathematica using f (x, y) = y − y/(x2 + y 2 ) and ContourPlot[f[x, y], {x, -3, 3}, {y, -3, 3}, Axes−>True, AxesOrigin−>{0, 0}, AxesLabel−>{x, y}, Frame−>False, PlotPoints−>100, ContourShading−>False, Contours−>{0, -0.2, 0.2, -0.4, 0.4, -0.6, 0.6, -0.8, 0.8}] The second graph uses 8 x=− y 3 − cy 2 − y c−y and x= 8 y 3 − cy 2 − y . c−y In this case the x-axis is vertical and the y-axis is horizontal. To obtain the third graph, we solve y − y/(x2 + y 2 ) = c for y in a CAS. This appears to give one real and two complex solutions. When graphed in Mathematica however, all three solutions contribute to the graph. This is because the solutions involve the square root of expressions containing c. For some values of c the expression is negative, causing an apparent complex solution to actually be real. 52 2.5 Solutions by Substitutions -3 -2 y 3 x 3 y 3 2 2 2 1 1 1 -1 1 2 3 x -1.5 -10.5 0.511.5y -3 -2 -1 1 -1 -1 -1 -2 -2 -2 -3 -3 -3 EXERCISES 2.5 Solutions by Substitutions 1. Letting y = ux we have (x − ux) dx + x(u dx + x du) = 0 dx + x du = 0 dx + du = 0 x ln |x| + u = c x ln |x| + y = cx. 2. Letting y = ux we have (x + ux) dx + x(u dx + x du) = 0 (1 + 2u) dx + x du = 0 dx du + =0 x 1 + 2u 1 ln |x| + ln |1 + 2u| = c 2 " y# x2 1 + 2 = c1 x x2 + 2xy = c1 . 53 2 3 2.5 Solutions by Substitutions 3. Letting x = vy we have vy(v dy + y dv) + (y − 2vy) dy = 0 * + vy 2 dv + y v 2 − 2v + 1 dy = 0 dy v dv + =0 2 (v − 1) y 1 ln |v − 1| − + ln |y| = c v−1 , , ,x , 1 ln ,, − 1,, − + ln y = c y x/y − 1 (x − y) ln |x − y| − y = c(x − y). 4. Letting x = vy we have y(v dy + y dv) − 2(vy + y) dy = 0 y dv − (v + 2) dy = 0 dv dy − =0 v+2 y ln |v + 2| − ln |y| = c , , ,x , ln ,, + 2,, − ln |y| = c y x + 2y = c1 y 2 . 5. Letting y = ux we have * + u2 x2 + ux2 dx − x2 (u dx + x du) = 0 u2 dx − x du = 0 dx du − 2 =0 x u 1 =c u x ln |x| + = c y ln |x| + y ln |x| + x = cy. 6. Letting y = ux and using partial fractions, we have * 2 2 + u x + ux2 dx + x2 (u dx + x du) = 0 * + x2 u2 + 2u dx + x3 du = 0 dx du + =0 x u(u + 2) ln |x| + 1 1 ln |u| − ln |u + 2| = c 2 2 x2 u = c1 u+2 # "y y +2 x2 = c1 x x x2 y = c1 (y + 2x). 54 2.5 Solutions by Substitutions 7. Letting y = ux we have (ux − x) dx − (ux + x)(u dx + x du) = 0 * 2 + u + 1 dx + x(u + 1) du = 0 dx u+1 + 2 du = 0 x u +1 + 1 * ln |x| + ln u2 + 1 + tan−1 u = c 2 & $ 2 y y ln x2 + 1 + 2 tan−1 = c1 x2 x * + y ln x2 + y 2 + 2 tan−1 = c1 . x 8. Letting y = ux we have (x + 3ux) dx − (3x + ux)(u dx + x du) = 0 * 2 + u − 1 dx + x(u + 3) du = 0 u+3 dx + du = 0 x (u − 1)(u + 1) ln |x| + 2 ln |u − 1| − ln |u + 1| = c x(u − 1)2 = c1 u+1 "y #2 # "y x − 1 = c1 +1 x x (y − x)2 = c1 (y + x). 9. Letting y = ux we have −ux dx + (x + √ u x)(u dx + x du) = 0 √ (x + x u ) du + xu3/2 dx = 0 & $ dx 1 −3/2 du + =0 u + u x 2 2 −2u−1/2 + ln |u| + ln |x| = c ' ln |y/x| + ln |x| = 2 x/y + c y(ln |y| − c)2 = 4x. 10. Letting y = ux we have " ux + # ' x2 − (ux)2 dx − x(udx + xdu) du = 0 ' x2 − u2 x2 dx − x2 du = 0 ' x 1 − u2 dx − x2 du = 0, (x > 0) du dx −√ =0 x 1 − u2 ln x − sin−1 u = c sin−1 u = ln x + c1 55 2.5 Solutions by Substitutions sin−1 y = ln x + c2 x y = sin(ln x + c2 ) x y = x sin(ln x + c2 ). See Problem 33 in this section for an analysis of the solution. 11. Letting y = ux we have * + x3 − u3 x3 dx + u2 x3 (u dx + x du) = 0 dx + u2 x du = 0 dx + u2 du = 0 x 1 ln |x| + u3 = c 3 3x3 ln |x| + y 3 = c1 x3 . Using y(1) = 2 we find c1 = 8. The solution of the initial-value problem is 3x3 ln |x| + y 3 = 8x3 . 12. Letting y = ux we have (x2 + 2u2 x2 )dx − ux2 (u dx + x du) = 0 x2 (1 + u2 )dx − ux3 du = 0 dx u du =0 − x 1 + u2 1 ln |x| − ln(1 + u2 ) = c 2 x2 = c1 1 + u2 x4 = c1 (x2 + y 2 ). Using y(−1) = 1 we find c1 = 1/2. The solution of the initial-value problem is 2x4 = y 2 + x2 . 13. Letting y = ux we have (x + uxeu ) dx − xeu (u dx + x du) = 0 dx − xeu du = 0 dx − eu du = 0 x ln |x| − eu = c ln |x| − ey/x = c. Using y(1) = 0 we find c = −1. The solution of the initial-value problem is ln |x| = ey/x − 1. 14. Letting x = vy we have y(v dy + y dv) + vy(ln vy − ln y − 1) dy = 0 y dv + v ln v dy = 0 dv dy + =0 v ln v y ln |ln |v|| + ln |y| = c , , ,x, y ln ,, ,, = c1 . y 56 2.5 Solutions by Substitutions , , ,x, Using y(1) = e we find c1 = −e. The solution of the initial-value problem is y ln ,, ,, = −e. y 15. From y ′ + 1 1 dw 3 3 y = y −2 and w = y 3 we obtain + w = . An integrating factor is x3 so that x3 w = x3 + c x x dx x x or y 3 = 1 + cx−3 . 16. From y ′ − y = ex y 2 and w = y −1 we obtain or y −1 = − 12 ex + ce−x . dw + w = −ex . An integrating factor is ex so that ex w = − 21 e2x + c dx 17. From y ′ + y = xy 4 and w = y −3 we obtain xe−3x + 13 e−3x + c or y −3 = x + 1 3 + ce3x . dw − 3w = −3x. An integrating factor is e−3x so that e−3x w = dx $ $ & & 1 1 dw 18. From y ′ − 1 + + 1+ y = y 2 and w = y −1 we obtain w = −1. An integrating factor is xex so that x dx x c 1 xex w = −xex + ex + c or y −1 = −1 + + e−x . x x 1 1 dw 1 1 19. From y ′ − y = − 2 y 2 and w = y −1 we obtain + w = 2 . An integrating factor is t so that tw = ln t + c t t dt t t 1 c t = ln t + c, we see that the solution can also be expressed in the or y −1 = ln t + . Writing this in the form t t y form et/y = c1 t. 2t dw 2t −2t 2 w= . An integrating factor is y= y 4 and w = y −3 we obtain − 3 (1 + t2 ) 3 (1 + t2 ) dt 1 + t2 1 + t2 * + 1 w 1 so that = + c or y −3 = 1 + c 1 + t2 . 2 2 2 1+t 1+t 1+t 20. From y ′ + 21. From y ′ − 2 3 dw 6 9 y = 2 y 4 and w = y −3 we obtain + w = − 2 . An integrating factor is x6 so that x x dx x x x6 w = − 59 x5 + c or y −3 = − 95 x−1 + cx−6 . If y(1) = 1 2 then c = 49 5 and y −3 = − 59 x−1 + 49 −6 . 5 x 3 3 dw + w = . An integrating factor is e3x/2 so that e3x/2 w = dx 2 2 = 1 + ce−3x/2 . If y(0) = 4 then c = 7 and y 3/2 = 1 + 7e−3x/2 . 22. From y ′ + y = y −1/2 and w = y 3/2 we obtain e3x/2 + c or y 3/2 du 1 − 1 = u2 or du = dx. Thus tan−1 u = x + c or dx 1 + u2 u = tan(x + c), and x + y + 1 = tan(x + c) or y = tan(x + c) − x − 1. 23. Let u = x + y + 1 so that du/dx = 1 + dy/dx. Then 24. Let u = x + y so that du/dx = 1 + dy/dx. Then and (x + y)2 = 2x + c1 . du 1−u −1 = or u du = dx. Thus 21 u2 = x + c or u2 = 2x + c1 , dx u 25. Let u = x + y so that du/dx = 1 + dy/dx. Then du − 1 = tan2 u or cos2 u du = dx. Thus 12 u + dx 1 4 sin 2u = x + c or 2u + sin 2u = 4x + c1 , and 2(x + y) + sin 2(x + y) = 4x + c1 or 2y + sin 2(x + y) = 2x + c1 . 26. Let u = x + y so that du/dx = 1 + dy/dx. Then (1 − sin u)/(1 − sin u) we have 1 du − 1 = sin u or du = dx. Multiplying by dx 1 + sin u 1 − sin u du = dx or (sec2 u − sec u tan u)du = dx. Thus tan u − sec u = x + c or cos2 u tan(x + y) − sec(x + y) = x + c. 57 2.5 Solutions by Substitutions √ √ du 1 27. Let u = y − 2x + 3 so that du/dx = dy/dx − 2. Then + 2 = 2 + u or √ du = dx. Thus 2 u = x + c and dx u √ 2 y − 2x + 3 = x + c. du 28. Let u = y − x + 5 so that du/dx = dy/dx − 1. Then + 1 = 1 + eu or e−u du = dx. Thus −e−u = x + c and dx −ey−x+5 = x + c. du 1 29. Let u = x + y so that du/dx = 1 + dy/dx. Then − 1 = cos u and du = dx. Now dx 1 + cos u 1 1 − cos u 1 − cos u = csc2 u − csc u cot u = = 1 + cos u 1 − cos2 u sin2 u . . so we have (csc2 u − csc u cot u)du = dx and − cot u + csc u = x + c. Thus − cot(x + y) + csc(x + y) = x + c. √ Setting x = 0 and y = π/4 we obtain c = 2 − 1. The solution is √ csc(x + y) − cot(x + y) = x + 2 − 1. 30. Let u = 3x + 2y so that du/dx = 3 + 2 dy/dx. Then long division du 2u 5u + 6 u+2 = 3+ = and du = dx. Now by dx u+2 u+2 5u + 6 1 4 u+2 = + 5u + 6 5 25u + 30 so we have and 1 5u + 4 25 - $ 4 1 + 5 25u + 30 & du = dx ln |25u + 30| = x + c. Thus 1 4 (3x + 2y) + ln |75x + 50y + 30| = x + c. 5 25 Setting x = −1 and y = −1 we obtain c = or 4 25 ln 95. The solution is 4 4 1 (3x + 2y) + ln |75x + 50y + 30| = x + ln 95 5 25 25 5y − 5x + 2 ln |75x + 50y + 30| = 2 ln 95. 31. We write the differential equation M (x, y)dx + N (x, y)dy = 0 as dy/dx = f (x, y) where f (x, y) = − M (x, y) . N (x, y) The function f (x, y) must necessarily be homogeneous of degree 0 when M and N are homogeneous of degree α. Since M is homogeneous of degree α, M (tx, ty) = tα M (x, y), and letting t = 1/x we have M (1, y/x) = Thus 1 M (x, y) xα or M (x, y) = xα M (1, y/x). "y# dy xα M (1, y/x) M (1, y/x) . = f (x, y) = − α =− =F dx x N (1, y/x) N (1, y/x) x 32. Rewrite (5x2 − 2y 2 )dx − xy dy = 0 as xy dy = 5x2 − 2y 2 dx and divide by xy, so that dy x y =5 −2 . dx y x 58 2.5 Solutions by Substitutions We then identify F "y# x =5 " y #−1 x −2 "y# x . 33. (a) By inspection y = x and y = −x are solutions of the differential equation and not members of the family y = x sin(ln x + c2 ). (b) Letting x = 5 and y = 0 in sin−1 (y/x) = ln x + c2 we get sin−1 0 = ln 5 + c y −1 or c = − ln 5. Then sin (y/x) = ln x − ln 5 = ln(x/5). Because the range of the arcsine function is [−π/2, π/2] we must have π x π − ≤ ln ≤ 2 5 2 x −π/2 e ≤ ≤ eπ/2 5 −π/2 5e ≤ x ≤ 5eπ/2 . 20 15 10 5 5 10 15 20 x The interval of definition of the solution is approximately [1.04, 24.05]. 34. As x → −∞, e6x → 0 and y → 2x + 3. Now write (1 + ce6x )/(1 − ce6x ) as (e−6x + c)/(e−6x − c). Then, as x → ∞, e−6x → 0 and y → 2x − 3. 35. (a) The substitutions y = y1 + u and dy dy1 du = + dx dx dx lead to du dy1 + = P + Q(y1 + u) + R(y1 + u)2 dx dx = P + Qy1 + Ry12 + Qu + 2y1 Ru + Ru2 or du − (Q + 2y1 R)u = Ru2 . dx This is a Bernoulli equation with n = 2 which can be reduced to the linear equation dw + (Q + 2y1 R)w = −R dx by the substitution w = u−1 . $ & 1 4 dw + − + w = −1. An integrating (b) Identify P (x) = −4/x , Q(x) = −1/x, and R(x) = 1. Then dx x x 0−1 / 2 factor is x3 so that x3 w = − 14 x4 + c or u = − 41 x + cx−3 . Thus, y = + u. x 36. Write the differential equation in the form x(y ′ /y) = ln x + ln y and let u = ln y. Then du/dx = y ′ /y and the differential equation becomes . x(du/dx) = ln x + u or du/dx − u/x = (ln x)/x, which is first-order and linear. 2 An integrating factor is e− dx/x The solution is = 1/x, so that (using integration by parts) u d 1 1 2 ln x 1 ln x u = 2 and =− − + c. dx x x x x x ln y = −1 − ln x + cx or y = 37. Write the differential equation as 1 dv + v = 32v −1 , dx x 59 ecx−1 . x 2.5 Solutions by Substitutions and let u = v 2 or v = u1/2 . Then 1 du dv = u−1/2 , dx 2 dx and substituting into the differential equation, we have 1 −1/2 du 1 1/2 = 32u−1/2 u + u 2 dx x or du 2 + u = 64. dx x . The latter differential equation is linear with integrating factor e (2/x)dx = x2 , so d 2 [x u] = 64x2 dx and x2 u = 64 3 x +c 3 or v2 = 64 c x+ 2 . 3 x 38. Write the differential equation as dP/dt − aP = −bP 2 and let u = P −1 or P = u−1 . Then dp du = −u−2 , dt dt and substituting into the differential equation, we have −u−2 du − au−1 = −bu−2 dt or . The latter differential equation is linear with integrating factor e du + au = b. dt a dt = eat , so d at [e u] = beat dt and b at e +c a b eat P −1 = eat + c a b P −1 = + ce−at a 1 a P = = . −at b/a + ce b + c1 e−at eat u = EXERCISES 2.6 A Numerical Method 1. We identify f (x, y) = 2x − 3y + 1. Then, for h = 0.1, yn+1 = yn + 0.1(2xn − 3yn + 1) = 0.2xn + 0.7yn + 0.1, and y(1.1) ≈ y1 = 0.2(1) + 0.7(5) + 0.1 = 3.8 For h = 0.05, y(1.2) ≈ y2 = 0.2(1.1) + 0.7(3.8) + 0.1 = 2.98. yn+1 = yn + 0.05(2xn − 3yn + 1) = 0.1xn + 0.85yn + 0.1, 60 2.6 A Numerical Method and y(1.05) ≈ y1 = 0.1(1) + 0.85(5) + 0.1 = 4.4 y(1.1) ≈ y2 = 0.1(1.05) + 0.85(4.4) + 0.1 = 3.895 y(1.15) ≈ y3 = 0.1(1.1) + 0.85(3.895) + 0.1 = 3.47075 y(1.2) ≈ y4 = 0.1(1.15) + 0.85(3.47075) + 0.1 = 3.11514. 2. We identify f (x, y) = x + y 2 . Then, for h = 0.1, yn+1 = yn + 0.1(xn + yn2 ) = 0.1xn + yn + 0.1yn2 , and y(0.1) ≈ y1 = 0.1(0) + 0 + 0.1(0)2 = 0 y(0.2) ≈ y2 = 0.1(0.1) + 0 + 0.1(0)2 = 0.01. For h = 0.05, yn+1 = yn + 0.05(xn + yn2 ) = 0.05xn + yn + 0.05yn2 , and y(0.05) ≈ y1 = 0.05(0) + 0 + 0.05(0)2 = 0 y(0.1) ≈ y2 = 0.05(0.05) + 0 + 0.05(0)2 = 0.0025 y(0.15) ≈ y3 = 0.05(0.1) + 0.0025 + 0.05(0.0025)2 = 0.0075 y(0.2) ≈ y4 = 0.05(0.15) + 0.0075 + 0.05(0.0075)2 = 0.0150. 3. Separating variables and integrating, we have dy = dx y and ln |y| = x + c. Thus y = c1 ex and, using y(0) = 1, we find c = 1, so y = ex is the solution of the initial-value problem. h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 h=0.05 yn 1.0000 1.1000 1.2100 1.3310 1.4641 1.6105 1.7716 1.9487 2.1436 2.3579 2.5937 Actual Value 1.0000 1.1052 1.2214 1.3499 1.4918 1.6487 1.8221 2.0138 2.2255 2.4596 2.7183 % Rel . Abs . Error Error 0.0000 0.00 0.0052 0.47 0.0114 0.93 0.0189 1.40 0.0277 1.86 0.0382 2.32 0.0506 2.77 0.0650 3.23 0.0820 3.68 0.1017 4.13 0.1245 4.58 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 yn 1.0000 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 1.4775 1.5513 1.6289 1.7103 1.7959 1.8856 1.9799 2.0789 2.1829 2.2920 2.4066 2.5270 2.6533 61 Actual Value 1.0000 1.0513 1.1052 1.1618 1.2214 1.2840 1.3499 1.4191 1.4918 1.5683 1.6487 1.7333 1.8221 1.9155 2.0138 2.1170 2.2255 2.3396 2.4596 2.5857 2.7183 % Rel . Abs . Error Error 0.0000 0.00 0.0013 0.12 0.0027 0.24 0.0042 0.36 0.0059 0.48 0.0077 0.60 0.0098 0.72 0.0120 0.84 0.0144 0.96 0.0170 1.08 0.0198 1.20 0.0229 1.32 0.0263 1.44 0.0299 1.56 0.0338 1.68 0.0381 1.80 0.0427 1.92 0.0476 2.04 0.0530 2.15 0.0588 2.27 0.0650 2.39 2.6 A Numerical Method 4. Separating variables and integrating, we have dy = 2x dx and y ln |y| = x2 + c. 2 2 Thus y = c1 ex and, using y(1) = 1, we find c = e−1 , so y = ex h=0.1 xn 1.00 1.10 1.20 1.30 1.40 1.50 5. 7. yn 1.0000 1.2000 1.4640 1.8154 2.2874 2.9278 Actual Value 1.0000 1.2337 1.5527 1.9937 2.6117 3.4903 Abs . Error 0.0000 0.0337 0.0887 0.1784 0.3243 0.5625 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 xn yn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 yn 0.0000 0.0500 0.0976 0.1429 0.1863 0.2278 0.2676 0.3058 0.3427 0.3782 0.4124 1.0000 1.1000 1.2155 1.3492 1.5044 1.6849 1.8955 2.1419 2.4311 2.7714 3.1733 yn 0.5000 0.5125 0.5232 0.5322 0.5395 0.5452 0.5496 0.5527 0.5547 0.5559 0.5565 % Rel . Abs . Error Error 0.0000 0.00 0.0079 0.72 0.0182 1.47 0.0314 2.27 0.0483 3.11 0.0702 4.00 0.0982 4.93 0.1343 5.90 0.1806 6.92 0.2403 7.98 0.3171 9.08 h=0.05 yn 1.0000 1.1000 1.2220 1.3753 1.5735 1.8371 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 62 Actual Value 1.0000 1.1079 1.2337 1.3806 1.5527 1.7551 1.9937 2.2762 2.6117 3.0117 3.4903 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 8. h=0.05 yn 0.5000 0.5250 0.5431 0.5548 0.5613 0.5639 % Rel . Error 0.00 2.73 5.71 8.95 12.42 16.12 6. h=0.05 yn 0.0000 0.1000 0.1905 0.2731 0.3492 0.4198 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 is the solution of the initial-value problem. h=0.05 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 −1 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 1.0000 1.0500 1.1053 1.1668 1.2360 1.3144 1.4039 1.5070 1.6267 1.7670 1.9332 h=0.05 yn 1.0000 1.1000 1.2159 1.3505 1.5072 1.6902 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 1.0000 1.0500 1.1039 1.1619 1.2245 1.2921 1.3651 1.4440 1.5293 1.6217 1.7219 2.6 9. h=0.1 10. h=0.05 xn 1.00 1.10 1.20 1.30 1.40 1.50 yn 1.0000 1.0000 1.0191 1.0588 1.1231 1.2194 xn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 h=0.1 yn 1.0000 1.0000 1.0049 1.0147 1.0298 1.0506 1.0775 1.1115 1.1538 1.2057 1.2696 xn 0.00 0.10 0.20 0.30 0.40 0.50 A Numerical Method h=0.05 yn 0.5000 0.5250 0.5499 0.5747 0.5991 0.6231 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 0.5000 0.5125 0.5250 0.5375 0.5499 0.5623 0.5746 0.5868 0.5989 0.6109 0.6228 11. Tables of values were computed using the Euler and RK4 methods. The resulting points were plotted and joined using ListPlot in Mathematica. h=0.25 h=0.1 y h=0.05 y 7 6 5 4 3 2 1 Euler 2 4 6 8 y 7 6 5 4 3 2 1 RK4 10 7 6 5 4 3 2 1 RK4 Euler x 2 4 6 8 10 RK4 Euler x 2 4 6 8 10 x 12. See the comments in Problem 11 above. h=0.25 h=0.1 y h=0.05 y 6 y 6 RK4 5 4 6 RK4 5 Euler 3 Euler 4 4 3 3 2 2 2 1 1 1 1 2 3 4 5 x 1 2 3 4 RK4 5 5 x Euler 1 2 3 4 5 x 13. Using separation of variables we find that the solution of the differential equation is y = 1/(1 − x2 ), which is undefined at x = 1, where the graph has a vertical asymptote. Because the actual solution of the differential equation becomes unbounded at x approaches 1, very small changes in the inputs x will result in large changes in the corresponding outputs y. This can be expected to have a serious effect on numerical procedures. The graphs below were obtained as described above in Problem 11. 63 2.6 A Numerical Method h=0.25 h=0.1 y 10 y 10 RK4 8 RK4 8 6 6 4 4 Euler 2 Euler 2 0.2 0.4 0.6 0.8 1 x 0.2 0.4 0.6 0.8 1 x EXERCISES 2.7 Linear Models 1. Let P = P (t) be the population at time t, and P0 the initial population. From dP/dt = kP we obtain P = P0 ekt . Using P (5) = 2P0 we find k = 1 5 ln 2 and P = P0 e(ln 2)t/5 . Setting P (t) = 3P0 we have 3 = e(ln 2)t/5 , so ln 3 = (ln 2)t 5 and t= 5 ln 3 ≈ 7.9 years. ln 2 Setting P (t) = 4P0 we have 4 = e(ln 2)t/5 , so ln 4 = (ln 2)t 5 and t ≈ 10 years. 2. From Problem 1 the growth constant is k = 51 ln 2. Then P = P0 e(1/5)(ln 2)t and 10,000 = P0 e(3/5) ln 2 . Solving for P0 we get P0 = 10,000e−(3/5) ln 2 = 6,597.5. Now P (10) = P0 e(1/5)(ln 2)(10) = 6,597.5e2 ln 2 = 4P0 = 26,390. The rate at which the population is growing is P ′ (10) = kP (10) = 1 (ln 2)26,390 = 3658 persons/year. 5 3. Let P = P (t) be the population at time t. Then dP/dt = kP and P = cekt . From P (0) = c = 500 we see that P = 500ekt . Since 15% of 500 is 75, we have P (10) = 500e10k = 575. Solving for k, we get 1 575 1 k = 10 ln 500 = 10 ln 1.15. When t = 30, P (30) = 500e(1/10)(ln 1.15)30 = 500e3 ln 1.15 = 760 years and P ′ (30) = kP (30) = 1 (ln 1.15)760 = 10.62 persons/year. 10 4. Let P = P (t) be bacteria population at time t and P0 the initial number. From dP/dt = kP we obtain P = P0 ekt . Using P (3) = 400 and P (10) = 2000 we find 400 = P0 e3k or ek = (400/P0 )1/3 . From P (10) = 2000 we then have 2000 = P0 e10k = P0 (400/P0 )10/3 , so $ &−3/7 2000 2000 −7/3 = P and P = ≈ 201. 0 0 40010/3 40010/3 64 2.7 Linear Models 5. Let A = A(t) be the amount of lead present at time t. From dA/dt = kA and A(0) = 1 we obtain A = ekt . Using A(3.3) = 1/2 we find k = 1 3.3 ln(1/2). When 90% of the lead has decayed, 0.1 grams will remain. Setting A(t) = 0.1 we have et(1/3.3) ln(1/2) = 0.1, so 1 t ln = ln 0.1 3.3 2 and t= 3.3 ln 0.1 ≈ 10.96 hours. ln(1/2) 6. Let A = A(t) be the amount present at time t. From dA/dt = kA and A(0) = 100 we obtain A = 100ekt . Using A(6) = 97 we find k = 16 ln 0.97. Then A(24) = 100e(1/6)(ln 0.97)24 = 100(0.97)4 ≈ 88.5 mg. 7. Setting A(t) = 50 in Problem 6 we obtain 50 = 100ekt , so kt = ln 1 2 and t= ln(1/2) ≈ 136.5 hours. (1/6) ln 0.97 8. (a) The solution of dA/dt = kA is A(t) = A0 ekt . Letting A = 1 2 A0 and solving for t we obtain the half-life T = −(ln 2)/k. (b) Since k = −(ln 2)/T we have A(t) = A0 e−(ln 2)t/T = A0 2−t/T . (c) Writing 81 A0 = A0 2−t/T as 2−3 = 2−t/T and solving for t we get t = 3T . Thus, an initial amount A0 will decay to 18 A0 in three half-lives. 9. Let I = I(t) be the intensity, t the thickness, and I(0) = I0 . If dI/dt = kI and I(3) = 0.25I0 , then I = I0 ekt , k = 31 ln 0.25, and I(15) = 0.00098I0 . 10. From dS/dt = rS we obtain S = S0 ert where S(0) = S0 . (a) If S0 = $5000 and r = 5.75% then S(5) = $6665.45. (b) If S(t) =$10,000 then t = 12 years. (c) S ≈ $6651.82 11. Assume that A = A0 ekt and k = −0.00012378. If A(t) = 0.145A0 then t ≈15,600 years. 12. From Example 3 in the text, the amount of carbon present at time t is A(t) = A0 e−0.00012378t . Letting t = 660 and solving for A0 we have A(660) = A0 e−0.0001237(660) = 0.921553A0 . Thus, approximately 92% of the original amount of C-14 remained in the cloth as of 1988. 13. Assume that dT /dt = k(T − 10) so that T = 10 + cekt . If T (0) = 70◦ and T (1/2) = 50◦ then c = 60 and k = 2 ln(2/3) so that T (1) = 36.67◦ . If T (t) = 15◦ then t = 3.06 minutes. 14. Assume that dT /dt = k(T − 5) so that T = 5 + cekt . If T (1) = 55◦ and T (5) = 30◦ then k = − 41 ln 2 and c = 59.4611 so that T (0) = 64.4611◦ . 15. Assume that dT /dt = k(T − 100) so that T = 100 + cekt . If T (0) = 20◦ and T (1) = 22◦ , then c = −80 and k = ln(39/40) so that T (t) = 90◦ , which implies t = 82.1 seconds. If T (t) = 98◦ then t = 145.7 seconds. 16. The differential equation for the first container is dT1 /dt = k1 (T1 − 0) = k1 T1 , whose solution is T1 (t) = c1 ek1 t . Since T1 (0) = 100 (the initial temperature of the metal bar), we have 100 = c1 and T1 (t) = 100ek1 t . After 1 minute, T1 (1) = 100ek1 = 90◦ C, so k1 = ln 0.9 and T1 (t) = 100et ln 0.9 . After 2 minutes, T1 (2) = 100e2 ln 0.9 = 100(0.9)2 = 81◦ C. The differential equation for the second container is dT2 /dt = k2 (T2 − 100), whose solution is T2 (t) = 100 + c2 ek2 t . When the metal bar is immersed in the second container, its initial temperature is T2 (0) = 81, so T2 (0) = 100 + c2 ek2 (0) = 100 + c2 = 81 65 2.7 Linear Models and c2 = −19. Thus, T2 (t) = 100 − 19ek2 t . After 1 minute in the second tank, the temperature of the metal bar is 91◦ C, so T2 (1) = 100 − 19ek2 = 91 9 ek2 = 19 9 k2 = ln 19 and T2 (t) = 100 − 19et ln(9/19) . Setting T2 (t) = 99.9 we have 100 − 19et ln(9/19) = 99.9 0.1 et ln(9/19) = 19 ln(0.1/19) t= ≈ 7.02. ln(9/19) Thus, from the start of the “double dipping” process, the total time until the bar reaches 99.9◦ C in the second container is approximately 9.02 minutes. 17. Using separation of variables to solve dT /dt = k(T − Tm ) we get T (t) = Tm + cekt . Using T (0) = 70 we find c = 70 − Tm , so T (t) = Tm + (70 − Tm )ekt . Using the given observations, we obtain T "1# = Tm + (70 − Tm )ek/2 = 110 2 T (1) = Tm + (70 − Tm )ek = 145. Then, from the first equation, ek/2 = (110 − Tm )/(70 − Tm ) and k k/2 2 e = (e ) = $ 110 − Tm 70 − Tm &2 = 145 − Tm 70 − Tm (110 − Tm )2 = 145 − Tm 70 − Tm 2 2 12100 − 220Tm + Tm = 10150 − 250Tm + Tm Tm = 390. The temperature in the oven is 390◦ . 18. (a) The initial temperature of the bath is Tm (0) = 60◦ , so in the short term the temperature of the chemical, which starts at 80◦ , should decrease or cool. Over time, the temperature of the bath will increase toward 100◦ since e−0.1t decreases from 1 toward 0 as t increases from 0. Thus, in the long term, the temperature of the chemical should increase or warm toward 100◦ . (b) Adapting the model for Newton’s law of cooling, we have dT = −0.1(T − 100 + 40e−0.1t ), dt T 100 T (0) = 80. 90 Writing the differential equation in the form 80 dT + 0.1T = 10 − 4e−0.1t dt . we see that it is linear with integrating factor e 70 0.1 dt 66 = e0.1t . 10 20 30 40 50 t 2.7 Linear Models Thus d 0.1t [e T ] = 10e0.1t − 4 dt e0.1t T = 100e0.1t − 4t + c and T (t) = 100 − 4te−0.1t + ce−0.1t . Now T (0) = 80 so 100 + c = 80, c = −20 and T (t) = 100 − 4te−0.1t − 20e−0.1t = 100 − (4t + 20)e−0.1t . The thinner curve verifies the prediction of cooling followed by warming toward 100◦ . The wider curve shows the temperature Tm of the liquid bath. 19. From dA/dt = 4 − A/50 we obtain A = 200 + ce−t/50 . A = 200 − 170e−t/50 . If A(0) = 30 then c = −170 and 20. From dA/dt = 0 − A/50 we obtain A = ce−t/50 . If A(0) = 30 then c = 30 and A = 30e−t/50 . 21. From dA/dt = 10 − A/100 we obtain A = 1000 + ce−t/100 . If A(0) = 0 then c = −1000 and A(t) = 1000 − 1000e−t/100 . 22. From Problem 21 the number of pounds of salt in the tank at time t is A(t) = 1000 − 1000e−t/100 . The concentration at time t is c(t) = A(t)/500 = 2 − 2e−t/100 . Therefore c(5) = 2 − 2e−1/20 = 0.0975 lb/gal and limt→∞ c(t) = 2. Solving c(t) = 1 = 2 − 2e−t/100 for t we obtain t = 100 ln 2 ≈ 69.3 min. 23. From dA 10A 2A = 10 − = 10 − dt 500 − (10 − 5)t 100 − t 1 we obtain A = 1000 − 10t + c(100 − t)2 . If A(0) = 0 then c = − 10 . The tank is empty in 100 minutes. 24. With cin (t) = 2 + sin(t/4) lb/gal, the initial-value problem is 1 t dA + A = 6 + 3 sin , A(0) = 50. dt 100 4 . The differential equation is linear with integrating factor e dt/100 = et/100 , so $ & d t/100 t A(t)] = 6 + 3 sin [e et/100 dt 4 150 t/100 t 3750 t/100 t et/100 A(t) = 600et/100 + e sin − e cos + c, 313 4 313 4 and 150 t 3750 t sin − cos + ce−t/100 . A(t) = 600 + 313 4 313 4 Letting t = 0 and A = 50 we have 600 − 3750/313 + c = 50 and c = −168400/313. Then A(t) = 600 + 150 t 3750 t 168400 −t/100 sin − cos − e . 313 4 313 4 313 The graphs on [0, 300] and [0, 600] below show the effect of the sine function in the input when compared with the graph in Figure 2.38(a) in the text. 67 2.7 Linear Models A !t" 600 A !t" 600 500 500 400 400 300 300 200 200 100 100 50 100 25. From 150 200 250 300 t 100 200 300 400 500 600 t dA 4A 2A =3− =3− dt 100 + (6 − 4)t 50 + t we obtain A = 50 + t + c(50 + t)−2 . If A(0) = 10 then c = −100,000 and A(30) = 64.38 pounds. 26. (a) Initially the tank contains 300 gallons of solution. Since brine is pumped in at a rate of 3 gal/min and the mixture is pumped out at a rate of 2 gal/min, the net change is an increase of 1 gal/min. Thus, in 100 minutes the tank will contain its capacity of 400 gallons. (b) The differential equation describing the amount of salt in the tank is A′ (t) = 6 − 2A/(300 + t) with solution A(t) = 600 + 2t − (4.95 × 107 )(300 + t)−2 , 0 ≤ t ≤ 100, as noted in the discussion following Example 5 in the text. Thus, the amount of salt in the tank when it overflows is A(100) = 800 − (4.95 × 107 )(400)−2 = 490.625 lbs. (c) When the tank is overflowing the amount of salt in the tank is governed by the differential equation $ & dA A = (3 gal/min)(2 lb/gal) − lb/gal (3 gal/min) dt 400 3A =6− , A(100) = 490.625. 400 Solving the equation, we obtain A(t) = 800 + ce−3t/400 . The initial condition yields c = −654.947, so that A(t) = 800 − 654.947e−3t/400 . When t = 150, A(150) = 587.37 lbs. (d) As t → ∞, the amount of salt is 800 lbs, which is to be expected since (400 gal)(2 lb/gal)= 800 lbs. (e) A 800 600 400 200 200 400 600 t 27. Assume L di/dt + Ri = E(t), L = 0.1, R = 50, and E(t) = 50 so that i = and limt→∞ i(t) = 3/5. 68 3 5 + ce−500t . If i(0) = 0 then c = −3/5 2.7 Linear Models 28. Assume L di/dt + Ri = E(t), E(t) = E0 sin ωt, and i(0) = i0 so that i= E0 R 2 L ω 2 + R2 sin ωt − E0 Lω 2 L ω 2 + R2 cos ωt + ce−Rt/L . E0 Lω . L2 ω 2 + R2 29. Assume R dq/dt + (1/C)q = E(t), R = 200, C = 10−4 , and E(t) = 100 so that q = 1/100 + ce−50t . If q(0) = 0 then c = −1/100 and i = 21 e−50t . Since i(0) = i0 we obtain c = i0 + 1 30. Assume R dq/dt + (1/C)q = E(t), R = 1000, C = 5 × 10−6 , and E(t) = 200. Then q = 1000 + ce−200t and 1 , q(0.005) = 0.003 coulombs, and i(0.005) = 0.1472 amps. We have i = −200ce−200t . If i(0) = 0.4 then c = − 500 1 q → 1000 as t → ∞. 31. For 0 ≤ t ≤ 20 the differential equation is 20 di/dt + 2i = 120. An integrating factor is et/10 , so (d/dt)[et/10 i] = 6et/10 and i = 60 + c1 e−t/10 . If i(0) = 0 then c1 = −60 and i = 60 − 60e−t/10 . For t > 20 the differential * + equation is 20 di/dt + 2i = 0 and i = c2 e−t/10 . At t = 20 we want c2 e−2 = 60 − 60e−2 so that c2 = 60 e2 − 1 . Thus i(t) = 3 32. Separating variables, we obtain 60 − 60e−t/10 , 0 ≤ t ≤ 20 * 2 + −t/10 60 e − 1 e , t > 20. dq dt = E0 − q/C k1 + k2 t , 1 q ,, , −C ln ,E0 − , = ln |k1 + k2 t| + c1 C k2 (E0 − q/C)−C = c2 . (k1 + k2 t)1/k2 1/k2 Setting q(0) = q0 we find c2 = (E0 − q0 /C)−C /k1 , so (E0 − q0 /C)−C (E0 − q/C)−C = 1/k 1/k (k1 + k2 t) 2 k1 2 &−1/k2 $ " k1 q0 #−C q #−C " = E0 − E0 − C C k + k2 t &1/Ck2 $ " # q k1 q0 E0 − = E0 − C C k + k2 t &1/Ck2 $ k1 . q = E0 C + (q0 − E0 C) k + k2 t 33. (a) From m dv/dt = mg − kv we obtain v = mg/k + ce−kt/m . If v(0) = v0 then c = v0 − mg/k and the solution of the initial-value problem is v(t) = (b) As t → ∞ the limiting velocity is mg/k. mg " mg # −kt/m . + v0 − e k k (c) From ds/dt = v and s(0) = 0 we obtain s(t) = mg mg # −kt/m m " mg # m" v0 − e + v0 − . t− k k k k k 34. (a) Integrating d2 s/dt2 = −g we get v(t) = ds/dt = −gt + c. From v(0) = 300 we find c = 300, and we are given g = 32, so the velocity is v(t) = −32t + 300. 69 2.7 Linear Models (b) Integrating again and using s(0) = 0 we get s(t) = −16t2 + 300t. The maximum height is attained when v = 0, that is, at ta = 9.375. The maximum height will be s(9.375) = 1406.25 ft. 35. When air resistance is proportional to velocity, the model for the velocity is m dv/dt = −mg − kv (using the fact that the positive direction is upward.) Solving the differential equation using separation of variables we obtain v(t) = −mg/k + ce−kt/m . From v(0) = 300 we get mg " mg # −kt/m v(t) = − + 300 + e . k k Integrating and using s(0) = 0 we find mg # mg m" s(t) = − 300 + (1 − e−kt/m ). t+ k k k Setting k = 0.0025, m = 16/32 = 0.5, and g = 32 we have s(t) = 1,340,000 − 6,400t − 1,340,000e−0.005t and v(t) = −6,400 + 6,700e−0.005t . The maximum height is attained when v = 0, that is, at ta = 9.162. The maximum height will be s(9.162) = 1363.79 ft, which is less than the maximum height in Problem 34. 36. Assuming that the air resistance is proportional to velocity and the positive direction is downward with s(0) = 0, the model for the velocity is m dv/dt = mg − kv. Using separation of variables to solve this differential equation, we obtain v(t) = mg/k + ce−kt/m . Then, using v(0) = 0, we get v(t) = (mg/k)(1 − e−kt/m ). Letting k = 0.5, m = (125 + 35)/32 = 5, and g = 32, we have v(t) = 320(1 − e−0.1t ). Integrating, we find s(t) = 320t + 3200e−0.1t + c1 . Solving s(0) = 0 for c1 we find c1 = −3200, therefore s(t) = 320t + 3200e−0.1t − 3200. At t = 15, when the parachute opens, v(15) = 248.598 and s(15) = 2314.02. At this time the value of k changes to k = 10 and the new initial velocity is v0 = 248.598. With the parachute open, the skydiver’s velocity is vp (t) = mg/k + c2 e−kt/m , where t is reset to 0 when the parachute opens. Letting m = 5, g = 32, and k = 10, this gives vp (t) = 16 + c2 e−2t . From v(0) = 248.598 we find c2 = 232.598, so vp (t) = 16 + 232.598e−2t . Integrating, we get sp (t) = 16t − 116.299e−2t + c3 . Solving sp (0) = 0 for c3 , we find c3 = 116.299, so sp (t) = 16t − 116.299e−2t + 116.299. Twenty seconds after leaving the plane is five seconds after the parachute opens. The skydiver’s velocity at this time is vp (5) = 16.0106 ft/s and she has fallen a total of s(15) + sp (5) = 2314.02 + 196.294 = 2510.31 ft. Her terminal velocity is limt→∞ vp (t) = 16, so she has very nearly reached her terminal velocity five seconds after the parachute opens. When the parachute opens, the distance to the ground is 15,000 − s(15) = 15,000 − 2,314 = 12,686 ft. Solving sp (t) = 12,686 we get t = 785.6 s = 13.1 min. Thus, it will take her approximately 13.1 minutes to reach the ground after her parachute has opened and a total of (785.6 + 15)/60 = 13.34 minutes after she exits the plane. . 37. (a) The differential equation is first-order and linear. Letting b = k/ρ, the integrating factor is e 3b dt/(bt+r0 ) = (r0 + bt)3 . Then d [(r0 + bt)3 v] = g(r0 + bt)3 dt and (r0 + bt)3 v = g (r0 + bt)4 + c. 4b The solution of the differential equation is v(t) = (g/4b)(r0 + bt) + c(r0 + bt)−3 . Using v(0) = 0 we find c = −gr04 /4b, so that v(t) = gr04 gρ " k # g gρr04 (r0 + bt) − = t − . r + 0 4b 4b(r0 + bt)3 4k ρ 4k(r0 + kt/ρ)3 (b) Integrating dr/dt = k/ρ we get r = kt/ρ + c. Using r(0) = r0 we have c = r0 , so r(t) = kt/ρ + r0 . 70 2.7 Linear Models (c) If r = 0.007 ft when t = 10 s, then solving r(10) = 0.007 for k/ρ, we obtain k/ρ = −0.0003 and r(t) = 0.01 − 0.0003t. Solving r(t) = 0 we get t = 33.3, so the raindrop will have evaporated completely at 33.3 seconds. 38. Separating variables, we obtain dP/P = k cos t dt, so ln |P | = k sin t + c and P = c1 ek sin t . If P (0) = P0 , then c1 = P0 and P = P0 ek sin t . 39. (a) From dP/dt = (k1 − k2 )P we obtain P = P0 e(k1 −k2 )t where P0 = P (0). (b) If k1 > k2 then P → ∞ as t → ∞. If k1 = k2 then P = P0 for every t. If k1 < k2 then P → 0 as t → ∞. 40. (a) Solving k1 (M − A) − k2 A = 0 for A we find the equilibrium solution A = k1 M/(k1 + k2 ). From the phase portrait we see that limt→∞ A(t) = k1 M/(k1 + k2 ). A Since k2 > 0, the material will never be completely memorized and the larger k2 is, the less the amount of material will be memorized over time. M k1 ################# k1 & k2 (b) Write the differential equation in the form dA/dt+(k1 +k2 )A = k1 M . Then an integrating factor is e(k1 +k2 )t , and d 1 (k1 +k2 )t 2 e A = k1 M e(k1 +k2 )t dt k1 M (k1 +k2 )t e(k1 +k2 )t A = e +c k1 + k2 A= Using A(0) = 0 we find c = − A→ k1 M . k1 + k2 k1 M + ce−(k1 +k2 )t . k1 + k2 # k1 M k1 M " and A = 1 − e−(k1 +k2 )t . As t → ∞, k1 + k2 k1 + k2 x 41. (a) Solving r −kx = 0 for x we find the equilibrium solution x = r/k. When x < r/k, dx/dt > 0 and when x > r/k, dx/dt < 0. From the phase portrait we see that limt→∞ x(t) = r/k. r #### k 71 2.7 Linear Models x (b) From dx/dt = r − kx and x(0) = 0 we obtain x = r/k − (r/k)e−kt so that x → r/k as t → ∞. If x(T ) = r/2k then T = (ln 2)/k. rêk t 42. The bar removed from the oven has an initial temperature of 300◦ F and, after being removed from the oven, approaches a temperature of 70◦ F. The bar taken from the room and placed in the oven has an initial temperature of 70◦ F and approaches a temperature of 300◦ F in the oven. Since the two temperature functions are continuous they must intersect at some time, t∗ . 43. (a) For 0 ≤ t < 4, 6 ≤ t < 10 and 12 ≤ t < 16, no voltage is applied to the heart and E(t) = 0. At the other times, the differential equation is dE/dt = −E/RC. Separating variables, integrating, and solving for e, (b) we get E = ke−t/RC , subject to E(4) = E(10) = E(16) = 12. These intitial conditions yield, respectively, k = 12e4/RC , k = 12e10/RC , k = 12e16/RC , and k = 12e22/RC . Thus ⎧ 0, 0 ≤ t < 4, 6 ≤ t < 10, 12 ≤ t < 16 ⎪ ⎪ ⎪ (4−t)/RC ⎪ 12e , 4 ≤t<6 ⎪ ⎨ (10−t)/RC , 10 ≤ t < 12 E(t) = 12e ⎪ ⎪ (16−t)/RC ⎪ 12e , 16 ≤ t < 18 ⎪ ⎪ ⎩ (22−t)/RC 12e , 22 ≤ t < 24. E 10 5 4 6 10 12 16 18 22 24 t 44. (a) (i) Using Newton’s second law of motion, F = ma = m dv/dt, the differential equation for the velocity v is m dv = mg sin θ dt or dv = g sin θ, dt where mg sin θ, 0 < θ < π/2, is the component of the weight along the plane in the direction of motion. (ii) The model now becomes dv m = mg sin θ − µmg cos θ, dt where µmg cos θ is the component of the force of sliding friction (which acts perpendicular to the plane) along the plane. The negative sign indicates that this component of force is a retarding force which acts in the direction opposite to that of motion. (iii) If air resistance is taken to be proportional to the instantaneous velocity of the body, the model becomes dv = mg sin θ − µmg cos θ − kv, dt where k is a constant of proportionality. m 72 2.7 Linear Models (b) (i) With m = 3 slugs, the differential equation is 3 dv 1 = (96) · dt 2 or dv = 16. dt Integrating the last equation gives v(t) = 16t + c1 . Since v(0) = 0, we have c1 = 0 and so v(t) = 16t. (ii) With m = 3 slugs, the differential equation is √ √ dv 1 3 3 = (96) · − · (96) · 3 dt 2 4 2 or dv = 4. dt In this case v(t) = 4t. (iii) When the retarding force due to air resistance is taken into account, the differential equation for velocity v becomes √ √ dv 3 3 1 1 dv 1 3 = (96) · − · (96) · − v or 3 = 12 − v. dt 2 4 2 4 dt 4 The last differential equation is linear and has solution v(t) = 48 + c1 e−t/12 . Since v(0) = 0, we find c1 = −48, so v(t) = 48 − 48e−t/12 . 45. (a) (i) If s(t) is distance measured down the plane from the highest point, then ds/dt = v. Integrating ds/dt = 16t gives s(t) = 8t2 + c2 . Using s(0) = 0 then gives c2 = 0. Now the length L of the plane is L = 50/ sin 30◦ = 100 ft. The time it takes the box to slide completely down the plane is the solution of s(t) = 100 or t2 = 25/2, so t ≈ 3.54 s. (ii) Integrating ds/dt = 4t gives s(t) = 2t2 + c2 . Using s(0) = 0 gives c2 = 0, so s(t) = 2t2 and the solution of s(t) = 100 is now t ≈ 7.07 s. (iii) Integrating ds/dt = 48 − 48e−t/12 and using s(0) = 0 to determine the constant of integration, we obtain s(t) = 48t + 576e−t/12 − 576. With the aid of a CAS we find that the solution of s(t) = 100, or 100 = 48t + 576e−t/12 − 576 or 0 = 48t + 576e−t/12 − 676, is now t ≈ 7.84 s. (b) The differential equation m dv/dt = mg sin θ − µmg cos θ can be written m dv = mg cos θ(tan θ − µ). dt If tan θ = µ, dv/dt = 0 and v(0) = 0 implies that v(t) = 0. If tan θ < µ and v(0) = 0, then integration implies v(t) = g cos θ(tan θ − µ)t < 0 for all time t. √ (c) Since tan 23◦ = 0.4245 and µ = 3/4 = 0.4330, we see that tan 23◦ < 0.4330. The differential equation √ is dv/dt = 32 cos 23◦ (tan 23◦ − 3/4) = −0.251493. Integration and the use of the initial condition gives v(t) = −0.251493t + 1. When the box stops, v(t) = 0 or 0 = −0.251493t + 1 or t = 3.976254 s. From s(t) = −0.125747t2 + t we find s(3.976254) = 1.988119 ft. (d) With v0 > 0, v(t) = −0.251493t + v0 and s(t) = −0.125747t2 + v0 t. Because two real positive solutions of the equation s(t) = 100, or 0 = −0.125747t2 + v0 t − 100, would be physically meaningless, we use the quadratic formula and require that b2 − 4ac = 0 or v02 − 50.2987 = 0. From this last equality we find v0 ≈ 7.092164 ft/s. For the time it takes the box to traverse the entire inclined plane, we must have 0 = −0.125747t2 + 7.092164t − 100. Mathematica gives complex roots for the last equation: t = 28.2001 ± 0.0124458i. But, for 0 = −0.125747t2 + 7.092164691t − 100, 73 2.7 Linear Models the roots are t = 28.1999 s and t = 28.2004 s. So if v0 > 7.092164, we are guaranteed that the box will slide completely down the plane. 46. (a) We saw in part (b) of Problem 34 that the ascent time is ta = 9.375. To find when the cannonball hits the ground we solve s(t) = −16t2 + 300t = 0, getting a total time in flight of t = 18.75 s. Thus, the time of descent is td = 18.75 − 9.375 = 9.375. The impact velocity is vi = v(18.75) = −300, which has the same magnitude as the initial velocity. (b) We saw in Problem 35 that the ascent time in the case of air resistance is ta = 9.162. Solving s(t) = 1,340,000 − 6,400t − 1,340,000e−0.005t = 0 we see that the total time of flight is 18.466 s. Thus, the descent time is td = 18.466 − 9.162 = 9.304. The impact velocity is vi = v(18.466) = −290.91, compared to an initial velocity of v0 = 300. EXERCISES 2.8 Nonlinear Models 1. (a) Solving N (1 − 0.0005N ) = 0 for N we find the equilibrium solutions N = 0 and N = 2000. N When 0 < N < 2000, dN/dt > 0. From the phase portrait we see that limt→∞ N (t) = 2000. A graph of the solution is shown in part (b). 2000 0 (b) Separating variables and integrating we have "1 # dN 1 = − dN = dt N (1 − 0.0005N ) N N − 2000 and N 2000 1500 1000 500 ln N − ln(N − 2000) = t + c. 5 10 15 20 t Solving for N we get N (t) = 2000ec+t /(1 + ec+t ) = 2000ec et /(1 + ec et ). Using N (0) = 1 and solving for ec we find ec = 1/1999 and so N (t) = 2000et /(1999 + et ). Then N (10) = 1833.59, so 1834 companies are expected to adopt the new technology when t = 10. 2. From dN/dt = N (a − bN ) and N (0) = 500 we obtain N= 500a . 500b + (a − 500b)e−at Since limt→∞ N = a/b = 50,000 and N (1) = 1000 we have a = 0.7033, b = 0.00014, and N = 50,000/(1 + 99e−0.7033t ) . 74 2.8 Nonlinear Models * + 3. From dP/dt = P 10−1 − 10−7 P and P (0) = 5000 we obtain P = 500/(0.0005 + 0.0995e−0.1t ) so that P → 1,000,000 as t → ∞. If P (t) = 500,000 then t = 52.9 months. 4. (a) We have dP/dt = P (a − bP ) with P (0) = 3.929 million. Using separation of variables we obtain 3.929a a/b = 3.929b + (a − 3.929b)e−at 1 + (a/3.929b − 1)e−at c , = 1 + (c/3.929 − 1)e−at P (t) = where c = a/b. At t = 60(1850) the population is 23.192 million, so 23.192 = c 1 + (c/3.929 − 1)e−60a or c = 23.192 + 23.192(c/3.929 − 1)e−60a . At t = 120(1910), 91.972 = c 1 + (c/3.929 − 1)e−120a or c = 91.972 + 91.972(c/3.929 − 1)(e−60a )2 . Combining the two equations for c we get $ (c − 23.192)/23.192 c/3.929 − 1 &2 " # c − 91.972 c −1 = 3.929 91.972 or 91.972(3.929)(c − 23.192)2 = (23.192)2 (c − 91.972)(c − 3.929). The solution of this quadratic equation is c = 197.274. This in turn gives a = 0.0313. Therefore, P (t) = (b) Year 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 Census Population 3.929 5.308 7.240 9.638 12.866 17.069 23.192 31.433 38.558 50.156 62.948 75.996 91.972 105.711 122.775 131.669 150.697 Predicted Population 3.929 5.334 7.222 9.746 13.090 17.475 23.143 30.341 39.272 50.044 62.600 76.666 91.739 107.143 122.140 136.068 148.445 Error 0.000 -0.026 0.018 -0.108 -0.224 -0.406 0.049 1.092 -0.714 0.112 0.348 -0.670 0.233 -1.432 0.635 -4.399 2.252 197.274 . 1 + 49.21e−0.0313t % Error 0.00 -0.49 0.24 -1.12 -1.74 -2.38 0.21 3.47 -1.85 0.22 0.55 -0.88 0.25 -1.35 0.52 -3.34 1.49 The model predicts a population of 159.0 million for 1960 and 167.8 million for 1970. The census populations for these years were 179.3 and 203.3, respectively. The percentage errors are 12.8 and 21.2, respectively. 75 2.8 Nonlinear Models P 5. (a) The differential equation is dP/dt = P (5 − P ) − 4. Solving P (5 − P ) − 4 = 0 for P we obtain equilibrium solutions P = 1 and P = 4. The phase portrait is shown on the right and solution curves are shown in part (b). We see that for P0 > 4 and 1 < P0 < 4 the population approaches 4 as t increases. For 0 < P < 1 the population decreases to 0 in finite time. 4 1 (b) The differential equation is P dP = P (5 − P ) − 4 = −(P 2 − 5P + 4) = −(P − 4)(P − 1). dt Separating variables and integrating, we obtain 4 1 dP = −dt (P − 4)(P − 1) & $ 1/3 1/3 − dP = −dt P −4 P −1 , , 1 ,, P − 4 ,, = −t + c ln , 3 P − 1, P −4 = c1 e−3t . P −1 3 t Setting t = 0 and P = P0 we find c1 = (P0 − 4)/(P0 − 1). Solving for P we obtain P (t) = 4(P0 − 1) − (P0 − 4)e−3t . (P0 − 1) − (P0 − 4)e−3t (c) To find when the population becomes extinct in the case 0 < P0 < 1 we set P = 0 in P0 − 4 −3t P −4 = e P −1 P0 − 1 from part (a) and solve for t. This gives the time of extinction 1 4(P0 − 1) t = − ln . 3 P0 − 4 5 5 6. Solving P (5 − P ) − 25 4 = 0 for P we obtain the equilibrium solution P = 2 . For P ̸= 2 , dP/dt < 0. Thus, if P0 < 52 , the population becomes extinct (otherwise there would be another equilibrium solution.) Using separation of variables to solve the initial-value problem, we get P (t) = [4P0 + (10P0 − 25)t]/[4 + (4P0 − 10)t]. To find when the population becomes extinct for P0 < 5 2 we solve P (t) = 0 for t. We see that the time of extinction is t = 4P0 /5(5 − 2P0 ). 7. Solving P (5 − P ) − 7 = 0 for P we obtain complex roots, so there are no equilibrium solutions. Since dP/dt < 0 for all values of P , the population becomes extinct for any initial condition. Using separation of variables to solve the initial-value problem, we get √ & √ ) ( $ 5 3 3 2P0 − 5 √ P (t) = + − tan tan−1 t . 2 2 2 3 76 2.8 Solving P (t) = 0 for t we see that the time of extinction is √ √ √ 0# / 2 "√ t= 3 tan−1 (5/ 3 ) + 3 tan−1 (2P0 − 5)/ 3 . 3 Nonlinear Models P 8. (a) The differential equation is dP/dt = P (1 − ln P ), which has the equilibrium solution P = e. When P0 > e, dP/dt < 0, and when P0 < e, dP/dt > 0. e t (b) The differential equation is dP/dt = P (1 + ln P ), which has the equilibrium P solution P = 1/e. When P0 > 1/e, dP/dt > 0, and when P0 < 1/e, dP/dt < 0. 1êe t (c) From dP/dt = P (a − b ln P ) we obtain −(1/b) ln |a − b ln P | = t + c1 so that P = ea/b e−ce −bt . If P (0) = P0 then c = (a/b) − ln P0 . 9. Let X = X(t) be the amount of C at time t and dX/dt = k(120 − 2X)(150 − X). If X(0) = 0 and X(5) = 10, then 150 − 150e180kt , 1 − 2.5e180kt where k = .0001259 and X(20) = 29.3 grams. Now by L’Hôpital’s rule, X → 60 as t → ∞, so that the amount X(t) = of A → 0 and the amount of B → 30 as t → ∞. 10. From dX/dt = k(150 − X)2 , X(0) = 0, and X(5) = 10 we obtain X = 150 − 150/(150kt + 1) where k = .000095238. Then X(20) = 33.3 grams and X → 150 as t → ∞ so that the amount of A → 0 and the amount of B → 0 as t → ∞. If X(t) = 75 then t = 70 minutes. √ h 11. (a) The initial-value problem is dh/dt = −8Ah h /Aw , h(0) = H. 10 8 Separating variables and integrating we have 6 √ 8A dh 8A √ = − h dt and 2 h = − h t + c. 4 Aw Aw h 2 √ Using h(0) = H we find c = 2 H , so the solution of the 500 ' √ initial-value problem is h(t) = (Aw H − 4Ah t)/Aw , where √ Aw H − 4Ah t ≥ 0. Thus, √ h(t) = (Aw H − 4Ah t)2 /A2w for 0 ≤ t ≤ Aw H/4Ah . 1000 1500 t ' (b) Identifying H = 10, Aw = 4π, and Ah = π/576 we have h(t) = t2 /331,776 − ( 5/2 /144)t + 10. Solving √ h(t) = 0 we see that the tank empties in 576 10 seconds or 30.36 minutes. 12. To obtain the solution of this differential equation we use h(t) from Problem 13 in Exercises 1.3. Then √ h(t) = (Aw H − 4cAh t)2 /A2w . Solving h(t) = 0 with c = 0.6 and the values from Problem 11 we see that the tank empties in 3035.79 seconds or 50.6 minutes. 77 2.8 Nonlinear Models 13. (a) Separating variables and integrating gives 6h3/2 dh = −5t and 12 5/2 h = −5t + c. 5 √ √ * +2/5 Using h(0) = 20 we find c = 1920 5 , so the solution of the initial-value problem is h(t) = 800 5− 25 t . 12 √ Solving h(t) = 0 we see that the tank empties in 384 5 seconds or 14.31 minutes. √ (b) When the height of the water is h, the radius of the top of the water is r = h tan 30◦ = h/ 3 and Aw = πh2 /3. The differential equation is dh π(2/12)2 √ 2 Ah ' 2gh = −0.6 64h = − 3/2 . = −c dt Aw πh2 /3 5h Separating variables and integrating gives 5h3/2 dh = −2 dt and 2h5/2 = −2t + c. Using h(0) = 9 we find c = 486, so the solution of the initial-value problem is h(t) = (243 − t)2/5 . Solving h(t) = 0 we see that the tank empties in 24.3 seconds or 4.05 minutes. 14. When the height of the water is h, the radius of the top of the water is 2 5 (20 2 Aw = 4π(20 − h) /25. The differential equation is − h) and √ √ Ah ' h dh π(2/12)2 5 . = −c 2gh = −0.6 64h = − dt Aw 4π(20 − h)2 /25 6 (20 − h)2 Separating variables and integrating we have √ (20 − h)2 5 2 5 80 √ dh = − dt and 800 h − h3/2 + h5/2 = − t + c. 6 3 5 6 h √ Using h(0) = 20 we find c = 2560 5/3, so an implicit solution of the initial-value problem is √ √ 2 5 80 2560 5 . 800 h − h3/2 + h5/2 = − t + 3 5 6 3 √ To find the time it takes the tank to empty we set h = 0 and solve for t. The tank empties in 1024 5 seconds or 38.16 minutes. Thus, the tank empties more slowly when the base of the cone is on the bottom. 15. (a) After separating variables we obtain m dv mg − kv 2 dv 1 √ g 1 − ( k v/√mg )2 ' √ mg k/mg dv √ √ √ k g 1 − ( k v/ mg )2 √ % m kv tanh−1 √ kg mg √ kv −1 tanh √ mg = dt = dt = dt =t+c = % kg t + c1 . m Thus the velocity at time t is : 9% mg kg tanh t + c1 . v(t) = k m √ √ Setting t = 0 and v = v0 we find c1 = tanh−1 ( k v0 / mg ). % 78 2.8 (b) Since tanh t → 1 as t → ∞, we have v → Nonlinear Models ' mg/k as t → ∞. . (c) Integrating the expression for v(t) in part (a) we obtain an integral of the form du/u: : :< 9% ; 9% % m mg kg kg + c2 . s(t) = tanh t + c1 dt = ln cosh t + c1 k m k m Setting t = 0 and s = 0 we find c2 = −(m/k) ln(cosh c1 ), where c1 is given in part (a). 16. The differential equation is m dv/dt = −mg − kv 2 . Separating variables and integrating, we have dv dt =− mg + kv 2 m 9√ : 1 1 kv √ =− t+c tan−1 √ mg m mgk 9√ : % kv gk −1 =− t + c1 tan √ mg m 9 : % % mg gk tan c1 − t . v(t) = k m Setting v(0) = 300, m = c1 = 0.914743. Integrating 16 32 = 1 2 , g = 32, and k = 0.0003, we find v(t) = 230.94 tan(c1 − 0.138564t) and v(t) = 230.94 tan(0.914743 − 0.138564t) we get s(t) = 1666.67 ln | cos(0.914743 − 0.138564t)| + c2 . Using s(0) = 0 we find c2 = 823.843. Solving v(t) = 0 we see that the maximum height is attained when t = 6.60159. The maximum height is s(6.60159) = 823.843 ft. 17. (a) Let ρ be the weight density of the water and V the volume of the object. Archimedes’ principle states that the upward buoyant force has magnitude equal to the weight of the water displaced. Taking the positive direction to be down, the differential equation is m dv = mg − kv 2 − ρV. dt (b) Using separation of variables we have m dv = dt (mg − ρV ) − kv 2 √ m k dv √ √ √ = dt k ( mg − ρV )2 − ( k v)2 √ m kv 1 −1 √ √ tanh √ = t + c. mg − ρV k mg − ρV Thus v(t) = % mg − ρV tanh k $√ (c) Since tanh t → 1 as t → ∞, the terminal velocity is & kmg − kρV t + c1 . m ' (mg − ρV )/k . 79 2.8 Nonlinear Models 18. (a) Writing the equation in the form (x − ' x2 + y 2 )dx + y dy = 0 we identify M = x − ' x2 + y 2 and N = y. Since M and N are both homogeneous functions of degree 1 we use the substitution y = ux. It follows that # " ' x − x2 + u2 x2 dx + ux(u dx + x du) = 0 2 1 ' x 1 − 1 + u2 + u2 dx + x2 u du = 0 u du dx √ = x 1 + u2 − 1 + u2 dx u du √ √ . = x 1 + u2 (1 − 1 + u2 ) √ √ Letting w = 1 − 1 + u2 we have dw = −u du/ 1 + u2 so that , , ' , , − ln ,1 − 1 + u2 , = ln |x| + c − 1 √ = c1 x 1 + u2 ' c2 1 − 1 + u2 = − x % c2 y2 1+ = 1+ 2 x x 1− 1+ (−c2 = 1/c1 ) y2 c2 2c2 + 22 = 1 + 2 . x x x Solving for y 2 we have "c # " c2 # 2 2 which is a family of parabolas symmetric with respect to the x-axis with vertex at (−c2 /2, 0) and focus at y 2 = 2c2 x + c22 = 4 2 x+ the origin. (b) Let u = x2 + y 2 so that du dy = 2x + 2y . dx dx Then 1 du dy = −x dx 2 dx and the differential equation can be written in the form y √ 1 du − x = −x + u 2 dx or 1 du √ = u. 2 dx Separating variables and integrating gives du √ = dx 2 u √ u=x+c u = x2 + 2cx + c2 x2 + y 2 = x2 + 2cx + c2 y 2 = 2cx + c2 . 19. (a) From 2W 2 − W 3 = W 2 (2 − W ) = 0 we see that W = 0 and W = 2 are constant solutions. 80 2.8 Nonlinear Models (b) Separating variables and using a CAS to integrate we get dW √ = dx W 4 − 2W and − tanh−1 "1√ 2 4 − 2W # = x + c. Using the facts that the hyperbolic tangent is an odd function and 1 − tanh2 x = sech2 x we have 1√ 4 − 2W = tanh(−x − c) = − tanh(x + c) 2 1 (4 − 2W ) = tanh2 (x + c) 4 1 1 − W = tanh2 (x + c) 2 1 W = 1 − tanh2 (x + c) = sech2 (x + c). 2 Thus, W (x) = 2 sech2 (x + c). (c) Letting x = 0 and W = 2 we find that sech2 (c) = 1 and c = 0. W 2 −3 3 x 20. (a) Solving r2 + (10 − h)2 = 102 for r2 we see that r2 = 20h − h2 . Combining the rate of input of water, π, with the rate of output due to evaporation, kπr2 = kπ(20h − h2 ), we have dV /dt = π − kπ(20h − h2 ). Using V = 10πh2 − 31 πh3 , we see also that dV /dt = (20πh − πh2 )dh/dt. Thus, (20πh − πh2 ) dh = π − kπ(20h − h2 ) dt and dh 1 − 20kh + kh2 = . dt 20h − h2 (b) Letting k = 1/100, separating variables and integrating (with the help of a CAS), we get 100h(h − 20) dh = dt (h − 10)2 and 100(h2 − 10h + 100) = t + c. 10 − h Using h(0) = 0 we find c = 1000, and solving for h we get h(t) = *√ + 0.005 t2 + 4000t−t , where the positive square root is chosen because h ≥ 0. h 10 8 6 4 2 t 2000 4000 6000 8000 10000 (c) The volume of the tank is V = 32 π(10)3 feet, so at a rate of π cubic feet per minute, the tank will fill in 2 3 3 (10) ≈ 666.67 minutes ≈ 11.11 hours. (d) At 666.67 minutes, the depth of the water is h(666.67) = 5.486 feet. From the graph in (b) we suspect that limt→∞ h(t) = 10, in which case the tank will never completely fill. To prove this we compute the limit of h(t): "' # t2 + 4000t − t2 lim h(t) = 0.005 lim t2 + 4000t − t = 0.005 lim √ t→∞ t→∞ t→∞ t2 + 4000t + t 4000 4000t = 0.005 lim ' = 0.005(2000) = 10. = 0.005 t→∞ t 1 + 4000/t + t 1+1 81 2.8 Nonlinear Models 21. (a) t P(t) Q(t) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 3.929 5.308 7.240 9.638 12.866 17.069 23.192 31.433 38.558 50.156 62.948 75.996 91.972 105.711 122.775 131.669 150.697 179.300 0.035 0.036 0.033 0.033 0.033 0.036 0.036 0.023 0.030 0.026 0.021 0.021 0.015 0.016 0.007 0.014 0.019 (b) The regression line is Q = 0.0348391 − 0.000168222P . Q 0.035 0.03 0.025 0.02 0.015 0.01 0.005 20 40 60 80 100 120 140 P (c) The solution of the logistic equation is given in equation (5) in the text. Identifying a = 0.0348391 and b = 0.000168222 we have P (t) = aP0 . bP0 + (a − bP0 )e−at (d) With P0 = 3.929 the solution becomes P (t) = (e) 0.136883 . 0.000660944 + 0.0341781e−0.0348391t P 175 150 125 100 75 50 25 25 50 75 100 125 150 t (f ) We identify t = 180 with 1970, t = 190 with 1980, and t = 200 with 1990. The model predicts P (180) = 188.661, P (190) = 193.735, and P (200) = 197.485. The actual population figures for these years are 203.303, 226.542, and 248.765 millions. As t → ∞, P (t) → a/b = 207.102. 82 2.8 Nonlinear Models 22. (a) Using a CAS to solve P (1 − P ) + 0.3e−P = 0 for P we see that P = 1.09216 is an equilibrium solution. (b) Since f (P ) > 0 for 0 < P < 1.09216, the solution P (t) of dP/dt = P (1 − P ) + 0.3e−P , f 2 P (0) = P0 , is increasing for P0 < 1.09216. Since f (P ) < 0 for P > 1.09216, the solution P (t) is decreasing for P0 > 1.09216. Thus P = 1.09216 is an attractor. 1 0.5 1 1.5 2 2.5 3 p -1 -2 (c) The curves for the second initial-value problem are thicker. The equilibrium solution for the logic model is P = 1. Comparing 1.09216 and 1, we see that the percentage increase is 9.216%. p 2 1.5 1 0.5 2 4 6 8 10 t 23. To find td we solve m dv = mg − kv 2 , dt v(0) = 0 using separation of variables. This gives v(t) = % mg tanh k % kg t. m Integrating and using s(0) = 0 gives 9 : % kg m ln cosh t . s(t) = k m To find the time of descent we solve s(t) = 823.84 and find td = 7.77882. The impact velocity is v(td ) = 182.998, which is positive because the positive direction is downward. ' 24. (a) Solving vt = mg/k for k we obtain k = mg/vt2 . The differential equation then becomes $ & dv dv mg 1 m = mg − 2 v 2 or = g 1 − 2 v2 . dt vt dt vt Separating variables and integrating gives vt tanh−1 v = gt + c1 . vt The initial condition v(0) = 0 implies c1 = 0, so v(t) = vt tanh gt . vt We find the distance by integrating: $ & gt v2 gt s(t) = vt tanh dt = t ln cosh + c2 . vt g vt 83 2.8 Nonlinear Models The initial condition s(0) = 0 implies c2 = 0, so s(t) = $ & gt vt2 ln cosh . g vt In 25 seconds she has fallen 20,000 − 14,800 = 5,200 feet. Using a CAS to solve & $ 32(25) 2 5200 = (vt /32) ln cosh vt for vt gives vt ≈ 271.711 ft/s. Then s(t) = & $ vt2 gt ln cosh = 2307.08 ln(cosh 0.117772t). g vt (b) At t = 15, s(15) = 2,542.94 ft and v(15) = s′ (15) = 256.287 ft/sec. 25. While the object is in the air its velocity is modeled by the linear differential equation m dv/dt = mg −kv. Using m = 160, k = 14 , and g = 32, the differential equation becomes dv/dt + (1/640)v = 32. The integrating factor . . is e dt/640 = et/640 and the solution of the differential equation is et/640 v = 32et/640 dt = 20,480et/640 + c. Using v(0) = 0 we see that c = −20,480 and v(t) = 20,480 − 20,480e−t/640 . Integrating we get s(t) = 20,480t + 13,107,200e−t/640 + c. Since s(0) = 0, c = −13,107,200 and s(t) = −13,107,200 + 20,480t + 13,107,200e−t/640 . To find when the object hits the liquid we solve s(t) = 500 − 75 = 425, obtaining ta = 5.16018. The velocity at the time of impact with the liquid is va = v(ta ) = 164.482. When the object is in the liquid its velocity is modeled by the nonlinear differential equation m dv/dt = mg − kv 2 . Using m = 160, g = 32, and k = 0.1 this becomes dv/dt = (51,200 − v 2 )/1600. Separating variables and integrating we have √ , √ , 2 ,, v − 160 2 ,, dv dt 1 √ = and ln t + c. = 51,200 − v 2 1600 640 , v + 160 2 , 1600 √ Solving v(0) = va = 164.482 we obtain c = −0.00407537. Then, for v < 160 2 = 226.274, √ √ , , √ √ , v − 160 2 , v − 160 2 2t/5−1.8443 , , √ = e 2t/5−1.8443 . or − , v + 160√2 ,= e v + 160 2 Solving for v we get v(t) = √ 2t/5 √ 2t/5 13964.6 − 2208.29e 61.7153 + 9.75937e Integrating we find √ s(t) = 226.275t − 1600 ln(6.3237 + e . 2t/5 ) + c. Solving s(0) = 0 we see that c = 3185.78, so √ s(t) = 3185.78 + 226.275t − 1600 ln(6.3237 + e 2t/5 ). To find when the object hits the bottom of the tank we solve s(t) = 75, obtaining tb = 0.466273. The time from when the object is dropped from the helicopter to when it hits the bottom of the tank is ta + tb = 5.62708 seconds. 84 2.9 Modeling with Systems of First-Order DEs EXERCISES 2.9 Modeling with Systems of First-Order DEs 1. The linear equation dx/dt = −λ1 x can be solved by either separation of variables or by an integrating factor. Integrating both sides of dx/x = −λ1 dt we obtain ln |x| = −λ1 t + c from which we get x = c1 e−λ1 t . Using x(0) = x0 we find c1 = x0 so that x = x0 e−λ1 t . Substituting this result into the second differential equation we have dy + λ2 y = λ1 x0 e−λ1 t dt which is linear. An integrating factor is eλ2 t so that d / λ2 t 0 e y = λ1 x0 e(λ2 −λ1 )t + c2 dt y= λ1 x0 −λ1 t λ1 x0 (λ2 −λ1 )t −λ2 t e e + c2 e−λ2 t = e + c2 e−λ2 t . λ2 − λ1 λ2 − λ1 Using y(0) = 0 we find c2 = −λ1 x0 /(λ2 − λ1 ). Thus y= + λ1 x0 * −λ1 t e − e−λ2 t . λ2 − λ1 Substituting this result into the third differential equation we have + λ1 λ2 x0 * −λ1 t dz e − e−λ2 t . = dt λ2 − λ1 Integrating we find z=− Using z(0) = 0 we find c3 = x0 . Thus λ2 x0 −λ1 t λ1 x0 −λ2 t e + e + c3 . λ2 − λ1 λ2 − λ1 $ z = x0 1 − & λ1 λ2 e−λ1 t + e−λ2 t . λ2 − λ1 λ 2 − λ1 2. We see from the graph that the half-life of A is approximately 4.7 days. To determine the half-life of B we use t = 50 as a base, since at this time the amount of substance A is so small that it contributes very little to substance B. Now we see from the graph that y(50) ≈ 16.2 and y(191) ≈ 8.1. Thus, the half-life of B is approximately 141 days. x, y, z 20 y(t) 15 10 5 x(t) z(t) 25 50 75 100 125 150 t 3. The amounts x and y are the same at about t = 5 days. The amounts x and z are the same at about t = 20 days. The amounts y and z are the same at about t = 147 days. The time when y and z are the same makes sense because most of A and half of B are gone, so half of C should have been formed. 4. Suppose that the series is described schematically by W =⇒ −λ1 X =⇒ −λ2 Y =⇒ −λ3 Z where −λ1 , −λ2 , and −λ3 are the decay constants for W , X and Y , respectively, and Z is a stable element. Let w(t), x(t), y(t), and 85 2.9 Modeling with Systems of First-Order DEs z(t) denote the amounts of substances W , X, Y , and Z, respectively. A model for the radioactive series is dw dt dx dt dy dt dz dt = −λ1 w = λ1 w − λ2 x = λ2 x − λ3 y = λ3 y. 5. The system is 1 1 2 1 x2 − x1 · 4 = − x1 + x2 + 6 50 50 25 50 1 1 1 2 2 x′2 = x1 · 4 − x2 − x2 · 3 = x1 − x2 . 50 50 50 25 25 x′1 = 2 · 3 + 6. Let x1 , x2 , and x3 be the amounts of salt in tanks A, B, and C, respectively, so that 1 x2 · 2 − 100 1 x′2 = x1 · 6 + 100 1 x′3 = x2 · 5 − 100 x′1 = 1 1 3 x1 · 6 = x2 − x1 100 50 50 1 1 1 3 7 1 x3 − x2 · 2 − x2 · 5 = x1 − x2 + x3 100 100 100 50 100 100 1 1 1 1 x3 − x3 · 4 = x2 − x3 . 100 100 20 20 7. (a) A model is dx1 x2 x1 =3· −2· , dt 100 − t 100 + t x1 x2 dx2 =2· −3· , dt 100 + t 100 − t x1 (0) = 100 x2 (0) = 50. (b) Since the system is closed, no salt enters or leaves the system and x1 (t) + x2 (t) = 100 + 50 = 150 for all time. Thus x1 = 150 − x2 and the second equation in part (a) becomes or dx2 2(150 − x2 ) 3x2 300 2x2 3x2 = − = − − dt 100 + t 100 − t 100 + t 100 + t 100 − t dx2 + dt $ 2 3 + 100 + t 100 − t which is linear in x2 . An integrating factor is & x2 = 300 , 100 + t e2 ln(100+t)−3 ln(100−t) = (100 + t)2 (100 − t)−3 so d [(100 + t)2 (100 − t)−3 x2 ] = 300(100 + t)(100 − t)−3 . dt Using integration by parts, we obtain ) ( 1 1 (100 + t)2 (100 − t)−3 x2 = 300 (100 + t)(100 − t)−2 − (100 − t)−1 + c . 2 2 Thus ) ( 300 1 1 2 3 (100 − t) + (100 + t)(100 − t) c(100 − t) − (100 + t)2 2 2 300 = [c(100 − t)3 + t(100 − t)]. (100 + t)2 x2 = 86 2.9 Modeling with Systems of First-Order DEs Using x2 (0) = 50 we find c = 5/3000. At t = 30, x2 = (300/1302 )(703 c + 30 · 70) ≈ 47.4 lbs. 8. A model is & $ dx1 1 = (4 gal/min)(0 lb/gal) − (4 gal/min) x1 lb/gal dt 200 $ & $ & dx2 1 1 = (4 gal/min) x1 lb/gal − (4 gal/min) x2 lb/gal dt 200 150 $ & $ & 1 1 dx3 = (4 gal/min) x2 lb/gal − (4 gal/min) x3 lb/gal dt 150 100 or dx1 1 = − x1 dt 50 dx2 1 2 = x1 − x2 dt 50 75 1 2 dx3 = x2 − x3 . dt 75 25 Over a long period of time we would expect x1 , x2 , and x3 to approach 0 because the entering pure water should flush the salt out of all three tanks. 9. Zooming in on the graph it can be seen that the populations are first equal at about t = 5.6. The approximate periods of x and y are both 45. x,y x 10 y 5 t 50 10. (a) The population y(t) approaches 10,000, while the population x(t) approaches extinction. 100 x,y 10 y 5 x (b) The population x(t) approaches 5,000, while the population y(t) approaches extinction. 10 20 10 20 10 20 t x,y 10 x 5 y (c) The population y(t) approaches 10,000, while the population x(t) x,y 10 approaches extinction. t y 5 x 87 t 2.9 Modeling with Systems of First-Order DEs (d) The population x(t) approaches 5,000, while the population y(t) x,y 10 approaches extinction. x 5 y 10 11. (a) x,y 10 y 5 x (b) (c) 40 t (d) y x 5 20 40 y x 20 x,y 10 t x,y 10 5 20 20 40 t x,y 10 y 5 x t 20 40 t In each case the population x(t) approaches 6,000, while the population y(t) approaches 8,000. 12. By Kirchhoff’s first law we have i1 = i2 + i3 . By Kirchhoff’s second law, on each loop we have E(t) = Li′1 + R1 i2 and E(t) = Li′1 + R2 i3 + q/C so that q = CR1 i2 − CR2 i3 . Then i3 = q ′ = CR1 i′2 − CR2 i3 so that the system is Li′2 + Li′3 + R1 i2 = E(t) 1 −R1 i′2 + R2 i′3 + i3 = 0. C 13. By Kirchhoff’s first law we have i1 = i2 + i3 . Applying Kirchhoff’s second law to each loop we obtain E(t) = i1 R1 + L1 di2 + i2 R2 dt E(t) = i1 R1 + L2 di3 + i3 R3 . dt and Combining the three equations, we obtain the system di2 + (R1 + R2 )i2 + R1 i3 = E dt di3 L2 + R1 i2 + (R1 + R3 )i3 = E. dt L1 14. By Kirchhoff’s first law we have i1 = i2 + i3 . By Kirchhoff’s second law, on each loop we have E(t) = Li′1 + Ri2 and E(t) = Li′1 + q/C so that q = CRi2 . Then i3 = q ′ = CRi′2 so that system is Li′ + Ri2 = E(t) CRi′2 + i2 − i1 = 0. 15. We first note that s(t) + i(t) + r(t) = n. Now the rate of change of the number of susceptible persons, s(t), is proportional to the number of contacts between the number of people infected and the number who are 88 2.9 Modeling with Systems of First-Order DEs susceptible; that is, ds/dt = −k1 si. We use −k1 < 0 because s(t) is decreasing. Next, the rate of change of the number of persons who have recovered is proportional to the number infected; that is, dr/dt = k2 i where k2 > 0 since r is increasing. Finally, to obtain di/dt we use d d (s + i + r) = n = 0. dt dt This gives di dr ds =− − = −k2 i + k1 si. dt dt dt The system of differential equations is then ds = −k1 si dt di = −k2 i + k1 si dt dr = k2 i. dt A reasonable set of initial conditions is i(0) = i0 , the number of infected people at time 0, s(0) = n − i0 , and r(0) = 0. 16. (a) If we know s(t) and i(t) then we can determine r(t) from s + i + r = n. (b) In this case the system is ds = −0.2si dt di = −0.7i + 0.2si. dt We also note that when i(0) = i0 , s(0) = 10 − i0 since r(0) = 0 and i(t) + s(t) + r(t) = 0 for all values of t. Now k2 /k1 = 0.7/0.2 = 3.5, so we consider initial conditions s(0) = 2, i(0) = 8; s(0) = 3.4, i(0) = 6.6; s(0) = 7, i(0) = 3; and s(0) = 9, i(0) = 1. 10 10 5 5 5 10 5 i s i i s,i s,i s,i s,i 10 i s s s 5 10 t 5 10 t 5 10 t 5 10 t We see that an initial susceptible population greater than k2 /k1 results in an epidemic in the sense that the number of infected persons increases to a maximum before decreasing to 0. On the other hand, when s(0) < k2 /k1 , the number of infected persons decreases from the start and there is no epidemic. 89 2.9 Modeling with Systems of First-Order DEs CHAPTER 2 REVIEW EXERCISES CHAPTER 2 REVIEW EXERCISES 1. Writing the differential equation in the form y ′ = k(y + A/k) we see that the critical point −A/k is a repeller for k > 0 and an attractor for k < 0. 2. Separating variables and integrating we have dy 4 = dx y x ln y = 4 ln x + c = ln x4 + c y = c1 x4 . We see that when x = 0, y = 0, so the initial-value problem has an infinite number of solutions for k = 0 and no solutions for k ̸= 0. dy 3. = (y − 1)2 (y − 3)2 dx 4. dy = y(y − 2)2 (y − 4) dx 5. When n is odd, xn < 0 for x < 0 and xn > 0 for x > 0. In this case 0 is unstable. When n is even, xn > 0 for x < 0 and for x > 0. In this case 0 is semi-stable. When n is odd, −xn > 0 for x < 0 and −xn < 0 for x > 0. In this case 0 is asymptotically stable. When n is even, −xn < 0 for x < 0 and for x > 0. In this case 0 is semi-stable. 6. Using a CAS we find that the zero of f occurs at approximately P = 1.3214. From the graph we observe that dP/dt > 0 for P < 1.3214 and dP/dt < 0 for P > 1.3214, so P = 1.3214 is an asymptotically stable critical point. Thus, limt→∞ P (t) = 1.3214. y 7. x 8. (a) linear in y, homogeneous, exact (b) linear in x (c) separable, exact, linear in x and y (d) Bernoulli in x (e) separable (f ) separable, linear in x, Bernoulli (g) linear in x (h) homogeneous 90 CHAPTER 2 REVIEW EXERCISES (i) Bernoulli (j) homogeneous, exact, Bernoulli (k) linear in x and y, exact, separable, homogeneous (l) exact, linear in y (m) homogeneous (n) separable 9. Separating variables and using the identity cos2 x = 12 (1 + cos 2x), we have y dy, +1 + 1 1 * 1 x + sin 2x = ln y 2 + 1 + c, 2 4 2 cos2 x dx = and y2 * + 2x + sin 2x = 2 ln y 2 + 1 + c. 10. Write the differential equation in the form x y ln dx = y $ & x x ln − y dy. y This is a homogeneous equation, so let x = uy. Then dx = u dy + y du and the differential equation becomes y ln u(u dy + y du) = (uy ln u − y) dy or y ln u du = −dy. Separating variables, we obtain ln u du = − dy y u ln |u| − u = − ln |y| + c , , x ,, x ,, x ln , , − = − ln |y| + c y y y x(ln x − ln y) − x = −y ln |y| + cy. 11. The differential equation 2 3x2 −2 dy + y=− y dx 6x + 1 6x + 1 is Bernoulli. Using w = y 3 , we obtain the linear equation dw 6 9x2 + w=− . dx 6x + 1 6x + 1 An integrating factor is 6x + 1, so d [(6x + 1)w] = −9x2 , dx 3x3 c w=− + , 6x + 1 6x + 1 and (6x + 1)y 3 = −3x3 + c. (Note: The differential equation is also exact.) 12. Write the differential equation in the form (3y 2 + 2x)dx + (4y 2 + 6xy)dy = 0. Letting M = 3y 2 + 2x and N = 4y 2 + 6xy we see that My = 6y = Nx , so the differential equation is exact. From fx = 3y 2 + 2x we obtain 91 CHAPTER 2 REVIEW EXERCISES f = 3xy 2 + x2 + h(y). Then fy = 6xy + h′ (y) = 4y 2 + 6xy and h′ (y) = 4y 2 so h(y) = 34 y 3 . A one-parameter family of solutions is 4 3xy 2 + x2 + y 3 = c. 3 13. Write the equation in the form dQ 1 + Q = t3 ln t. dt t An integrating factor is eln t = t, so d [tQ] = t4 ln t dt 1 1 tQ = − t5 + t5 ln t + c 25 5 and Q=− 1 4 1 4 c t + t ln t + . 25 5 t 14. Letting u = 2x + y + 1 we have du =2+ dx and so the given differential equation is transformed into & $ du − 2 = 1 or u dx dy , dx du 2u + 1 = . dx u Separating variables and integrating we get u du = dx 2u + 1 $ & 1 1 1 du = dx − 2 2 2u + 1 1 1 u − ln |2u + 1| = x + c 2 4 2u − ln |2u + 1| = 2x + c1 . Resubstituting for u gives the solution or 4x + 2y + 2 − ln |4x + 2y + 3| = 2x + c1 2x + 2y + 2 − ln |4x + 2y + 3| = c1 . 15. Write the equation in the form dy 8x 2x + 2 y= 2 . dx x + 4 x +4 * +4 An integrating factor is x2 + 4 , so and +4 2 * +3 d 1* 2 x + 4 y = 2x x2 + 4 dx * 2 +4 +4 1* 2 x +4 y = x +4 +c 4 y= * +−4 1 . + c x2 + 4 4 92 CHAPTER 2 REVIEW EXERCISES 16. Letting M = 2r2 cos θ sin θ + r cos θ and N = 4r + sin θ − 2r cos2 θ we see that Mr = 4r cos θ sin θ + cos θ = Nθ , so the differential equation is exact. From fθ = 2r2 cos θ sin θ + r cos θ we obtain f = −r2 cos2 θ + r sin θ + h(r). Then fr = −2r cos2 θ + sin θ + h′ (r) = 4r + sin θ − 2r cos2 θ and h′ (r) = 4r so h(r) = 2r2 . The solution is −r2 cos2 θ + r sin θ + 2r2 = c. 17. The differential equation has the form (d/dx) [(sin x)y] = 0. Integrating, we have (sin x)y = c or y = c/ sin x. The initial condition implies c = −2 sin(7π/6) = 1. Thus, y = 1/ sin x, where the interval π < x < 2π is chosen to include x = 7π/6. 18. Separating variables and integrating we have dy = −2(t + 1) dt y2 1 − = −(t + 1)2 + c y 1 y= , (t + 1)2 + c1 where −c = c1 . The initial condition y(0) = − 81 implies c1 = −9, so a solution of the initial-value problem is y= 1 (t + 1)2 − 9 or y= t2 1 , + 2t − 8 where −4 < t < 2. √ 19. (a) For y < 0, y is not a real number. (b) Separating variables and integrating we have dy √ = dx y and √ 2 y = x + c. √ Letting y(x0 ) = y0 we get c = 2 y0 − x0 , so that √ √ 2 y = x + 2 y0 − x0 and y = 1 √ (x + 2 y0 − x0 )2 . 4 √ √ y > 0 for y ̸= 0, we see that dy/dx = 21 (x + 2 y0 − x0 ) must be positive. Thus, the interval on √ which the solution is defined is (x0 − 2 y0 , ∞). Since 20. (a) The differential equation is homogeneous and we let y = ux. Then (x2 − y 2 ) dx + xy dy = 0 (x2 − u2 x2 ) dx + ux2 (u dx + x du) = 0 dx + ux du = 0 u du = − dx x 1 2 u = − ln |x| + c 2 y2 = −2 ln |x| + c1 . x2 The initial condition gives c1 = 2, so an implicit solution is y 2 = x2 (2 − 2 ln |x|). 93 CHAPTER 2 REVIEW EXERCISES (b) Solving for y in part (a) and being sure that the initial condition is √ still satisfied, we have y = − 2 |x|(1 − ln |x|)1/2 , where −e ≤ x ≤ e so that 1 − ln |x| ≥ 0. The graph of this function indicates that the derivative is not defined at x = 0 and x = e. Thus, √ the solution of the initial-value problem is y = − 2 x(1 − ln x)1/2 , for y 2 1 -2 0 < x < e. -1 1 2 x -1 -2 21. The graph of y1 (x) is the portion of the closed black curve lying in the fourth quadrant. Its interval of definition is approximately (0.7, 4.3). The graph of y2 (x) is the portion of the left-hand black curve lying in the third quadrant. Its interval of definition is (−∞, 0). 22. The first step of Euler’s method gives y(1.1) ≈ 9 + 0.1(1 + 3) = 9.4. Applying Euler’s method one more time √ gives y(1.2) ≈ 9.4 + 0.1(1 + 1.1 9.4 ) ≈ 9.8373. 23. From dP = 0.018P and P (0) = 4 billion we obtain P = 4e0.018t so that P (45) = 8.99 billion. dt 24. Let A = A(t) be the volume of CO2 at time t. From dA/dt = 1.2 − A/4 and A(0) = 16 ft3 we obtain A = 4.8 + 11.2e−t/4 . Since A(10) = 5.7 ft3 , the concentration is 0.017%. As t → ∞ we have A → 4.8 ft3 or 0.06%. 25. Separating variables, we have ' s2 − y 2 dy = −dx. y Substituting y = s sin θ, this becomes ' s2 − s2 sin2 θ (s cos θ)dθ s sin θ cos2 s dθ sin θ 1 − sin2 θ s dθ sin θ s (csc θ − sin θ)dθ Letting s = 10, this is = −dx = − dx = −x + c = −x + c s ln | csc θ − cot θ| + s cos θ = −x + c , , ' , s 's2 − y 2 , s2 − y 2 , , = −x + c. s ln , − ,+s , ,y y s , , , 10 '100 − y 2 , ' , , 10 ln , − , + 100 − y 2 = −x + c. , ,y y Letting x = 0 and y = 10 we determine that c = 0, so the solution is , , , 10 '100 − y 2 , ' , , 10 ln , − , + 100 − y 2 = −x. ,y , y 26. From V dC/dt = kA(Cs − C) and C(0) = C0 we obtain C = Cs + (C0 − Cs )e−kAt/V . 94 CHAPTER 2 REVIEW EXERCISES 27. (a) The differential equation dT = k(T − Tm ) = k[T − T2 − B(T1 − T )] dt $ BT1 + T2 = k[(1 + B)T − (BT1 + T2 )] = k(1 + B) T − 1+B & is autonomous and has the single critical point (BT1 + T2 )/(1 + B). Since k < 0 and B > 0, by phase-line analysis it is found that the critical point is an attractor and lim T (t) = t→∞ BT1 + T2 . 1+B Moreover, & $ BT1 + T2 BT1 + T2 = . lim Tm (t) = lim [T2 + B(T1 − T )] = T2 + B T1 − t→∞ t→∞ 1+B 1+B (b) The differential equation is or dT = k(T − Tm ) = k(T − T2 − BT1 + BT ) dt dT − k(1 + B)T = −k(BT1 + T2 ). dt . This is linear and has integrating factor e− k(1+B)dt = e−k(1+B)t . Thus, d −k(1+B)t [e T ] = −k(BT1 + T2 )e−k(1+B)t dt BT1 + T2 −k(1+B)t e−k(1+B)t T = e +c 1+B BT1 + T2 + cek(1+B)t . T (t) = 1+B Since k is negative, limt→∞ T (t) = (BT1 + T2 )/(1 + B). (c) The temperature T (t) decreases to (BT1 + T2 )/(1 + B), whereas Tm (t) increases to (BT1 + T2 )/(1 + B) as t → ∞. Thus, the temperature (BT1 + T2 )/(1 + B), (which is a weighted average, B 1 T1 + T2 , 1+B 1+B of the two initial temperatures), can be interpreted as an equilibrium temperature. The body cannot get cooler than this value whereas the medium cannot get hotter than this value. 28. (a) By separation of variables and partial fractions, , , $ & ,T − T , T , m, −1 3 ln , = 4Tm kt + c. , − 2 tan , T + Tm , Tm Then rewrite the right-hand side of the differential equation as dT 4 4 = k(T 4 − Tm ) = [(Tm + (T − Tm ))4 − Tm ] dt ;$ < &4 T − Tm 4 1+ = kTm −1 Tm ;9 < $ &2 : T − T T − T m m 4 1+4 = kTm · · · − 1 ← binomial expansion +6 Tm Tm 95 CHAPTER 2 REVIEW EXERCISES (b) When T − Tm is small compared to Tm , every term in the expansion after the first two can be ignored, giving dT ≈ k1 (T − Tm ), dt where 3 k1 = 4kTm . 29. We first solve (1 − t/10)di/dt + 0.2i = 4. Separating variables we obtain di/(40 − 2i) = dt/(10 − t). Then √ 1 − ln |40 − 2i| = − ln |10 − t| + c or 40 − 2i = c1 (10 − t). 2 √ Since i(0) = 0 we must have c1 = 2/ 10 . Solving for i we get i(t) = 4t − 15 t2 , 0 ≤ t < 10. For t ≥ 10 the equation for the current becomes 0.2i = 4 or i = 20. Thus ! 4t − 15 t2 , 0 ≤ t < 10 i(t) = 20, t ≥ 10. 20 10 10 20 The graph of i(t) is given in the figure. / 0 √ √ 30. From y 1 + (y ′ )2 = k we obtain dx = ( y/ k − y )dy. If y = k sin2 θ then & $ k 1 1 − cos 2θ dθ, and x = kθ − sin 2θ + c. dy = 2k sin θ cos θ dθ, dx = 2k 2 2 2 If x = 0 when θ = 0 then c = 0. 1 1 2 31. Letting c = 0.6, Ah = π( 32 · 12 ) , Aw = π · 12 = π, and g = 32, the differential equation becomes √ √ dh/dt = −0.00003255 h . Separating variables and integrating, we get 2 h = −0.00003255t + c, so h = √ √ (c1 − 0.00001628t)2 . Setting h(0) = 2, we find c = 2 , so h(t) = ( 2 − 0.00001628t)2 , where h is measured in feet and t in seconds. 32. One hour is 3,600 seconds, so the hour mark should be placed at √ h(3600) = [ 2 − 0.00001628(3600)]2 ≈ 1.838 ft ≈ 22.0525 in. up from the bottom of the tank. The remaining marks corresponding to the passage of 2, 3, 4, . . . , 12 hours are placed at the values shown in the table. The marks are not evenly spaced because the water is not draining out at a uniform rate; that is, h(t) is not a linear function of time. 33. In this case Aw = πh2 /4 and the differential equation is dh 1 =− h−3/2 . dt 7680 Separating variables and integrating, we have 1 dt 7680 1 =− t + c1 . 7680 h3/2 dh = − 2 5/2 h 5 96 time !seconds " 0 1 2 3 4 5 6 7 8 9 10 11 12 height !inches " 24.0000 22.0520 20.1864 18.4033 16.7026 15.0844 13.5485 12.0952 10.7242 9.4357 8.2297 7.1060 6.0648 CHAPTER 2 REVIEW EXERCISES √ Setting h(0) = 2 we find c1 = 8 2/5, so that √ 1 8 2 2 5/2 h =− t+ , 5 7680 5 √ 1 h5/2 = 4 2 − t, 3072 $ &2/5 √ 1 . h= 4 2− t 3072 and In this case h(4 hr) = h(14,400 s) = 11.8515 inches and h(5 hr) = h(18,000 s) is not a real number. Using a CAS to solve h(t) = 0, we see that the tank runs dry at t ≈ 17,378 s ≈ 4.83 hr. Thus, this particular conical water clock can only measure time intervals of less than 4.83 hours. 34. If we let rh denote the radius of the hole and Aw = π[f (h)]2 , then the √ √ differential equation dh/dt = −k h, where k = cAh 2g/Aw , becomes √ √ cπrh2 2g √ dh 8crh2 h =− . h=− dt π[f (h)]2 [f (h)]2 h 2 1 For the time marks to be equally spaced, the rate of change of the height must be a constant; that is, dh/dt = −a. (The constant is negative because the height is decreasing.) Thus √ 8crh2 h [f (h)] = , a √ 8crh2 h −a = − , [f (h)]2 2 and r = f (h) = 2rh −1 % 1 r 2c 1/4 h . a Solving for h, we have h= a2 r4 . 64c2 rh4 The shape of the tank with c = 0.6, a = 2 ft/12 hr = 1 ft/21,600 s, and rh = 1/32(12) = 1/384 is shown in the above figure. 35. From dx/dt = k1 x(α − x) we obtain $ 1/α 1/α + x α−x & dx = k1 dt so that x = αc1 eαk1 t /(1 + c1 eαk1 t ). From dy/dt = k2 xy we obtain , * +k2 /k1 k2 ,, ln 1 + c1 eαk1 t , + c or y = c2 1 + c1 eαk1 t . ln |y| = k1 36. In tank A the salt input is $ & &$ & $ &$ & $ gal lb x2 lb lb gal 1 7 x2 . 2 + 1 = 14 + min gal min 100 gal 100 min The salt output is $ 3 gal min &$ x1 lb 100 gal & $ &$ & x1 lb gal 2 lb + 5 = x1 . min 100 gal 25 min $ &$ In tank B the salt input is The salt output is $ 1 gal min &$ 5 gal min x2 lb 100 gal & x1 lb 100 gal & = 1 lb x1 . 20 min $ &$ & x2 lb lb gal 1 + 4 = x2 . min 100 gal 20 min 97 CHAPTER 2 REVIEW EXERCISES The system of differential equations is then dx1 1 2 = 14 + x2 − x1 dt 100 25 dx2 1 1 = x1 − x2 . dt 20 20 37. From y = −x − 1 + c1 ex we obtain y ′ = y + x so that the differential equation of the orthogonal family is 1 dy =− dx y+x dx + x = −y. dy . This is a linear differential equation and has integrating factor e dy = ey , so or d y [e x] = −yey dy ey x = −yey + ey + c2 x = −y + 1 + c2 e−y . 38. Differentiating the family of curves, we have y 5 1 1 y′ = − =− 2. 2 (x + c1 ) y The differential equation for the family of orthogonal trajectories is then y ′ = y 2 . Separating variables and integrating we get -5 dy = dx y2 1 − = x + c1 y 1 y=− . x + c1 5 x -5 98 3 Higher-Order Differential Equations EXERCISES 3.1 Preliminary Theory: Linear Equations 1. From y = c1 ex + c2 e−x we find y ′ = c1 ex − c2 e−x . Then y(0) = c1 + c2 = 0, y ′ (0) = c1 − c2 = 1 so that c1 = 1 2 and c2 = − 21 . The solution is y = 12 ex − 21 e−x . 2. From y = c1 e4x + c2 e−x we find y ′ = 4c1 e4x − c2 e−x . Then y(0) = c1 + c2 = 1, y ′ (0) = 4c1 − c2 = 2 so that c1 = 53 and c2 = 25 . The solution is y = 53 e4x + 52 e−x . 3. From y = c1 x + c2 x ln x we find y ′ = c1 + c2 (1 + ln x). Then y(1) = c1 = 3, y ′ (1) = c1 + c2 = −1 so that c1 = 3 and c2 = −4. The solution is y = 3x − 4x ln x. 4. From y = c1 + c2 cos x + c3 sin x we find y ′ = −c2 sin x + c3 cos x and y ′′ = −c2 cos x − c3 sin x. Then y(π) = c1 − c2 = 0, y ′ (π) = −c3 = 2, y ′′ (π) = c2 = −1 so that c1 = −1, c2 = −1, and c3 = −2. The solution is y = −1 − cos x − 2 sin x. 5. From y = c1 + c2 x2 we find y ′ = 2c2 x. Then y(0) = c1 = 0, y ′ (0) = 2c2 · 0 = 0 and hence y ′ (0) = 1 is not possible. Since a2 (x) = x is 0 at x = 0, Theorem 3.1 is not violated. 6. In this case we have y(0) = c1 = 0, y ′ (0) = 2c2 · 0 = 0 so c1 = 0 and c2 is arbitrary. Two solutions are y = x2 and y = 2x2 . 7. From x(0) = x0 = c1 we see that x(t) = x0 cos ωt + c2 sin ωt and x′ (t) = −x0 sin ωt + c2 ω cos ωt. Then x′ (0) = x1 = c2 ω implies c2 = x1 /ω. Thus x1 x(t) = x0 cos ωt + sin ωt. ω 8. Solving the system x(t0 ) = c1 cos ωt0 + c2 sin ωt0 = x0 x′ (t0 ) = −c1 ω sin ωt0 + c2 ω cos ωt0 = x1 for c1 and c2 gives c1 = Thus ωx0 cos ωt0 − x1 sin ωt0 ω and c2 = x1 cos ωt0 + ωx0 sin ωt0 . ω ωx0 cos ωt0 − x1 sin ωt0 x1 cos ωt0 + ωx0 sin ωt0 cos ωt + sin ωt ω ω x1 = x0 (cos ωt cos ωt0 + sin ωt sin ωt0 ) + (sin ωt cos ωt0 − cos ωt sin ωt0 ) ω x1 = x0 cos ω(t − t0 ) + sin ω(t − t0 ). ω x(t) = 9. Since a2 (x) = x − 2 and x0 = 0 the problem has a unique solution for −∞ < x < 2. 99 3.1 Preliminary Theory: Linear Equations 10. Since a0 (x) = tan x and x0 = 0 the problem has a unique solution for −π/2 < x < π/2. ! " ! " 11. (a) We have y(0) = c1 + c2 = 0, y ′′ (1) = c1 e + c2 e−1 = 1 so that c1 = e/ e2 − 1 and c2 = −e/ e2 − 1 . The ! " solution is y = e (ex − e−x ) / e2 − 1 . (b) We have y(0) = c3 cosh 0 + c4 sinh 0 = c3 = 0 and y(1) = c3 cosh 1 + c4 sinh 1 = c4 sinh 1 = 1, so c3 = 0 and c4 = 1/ sinh 1. The solution is y = (sinh x)/(sinh 1). (c) Starting with the solution in part (b) we have y= 1 2 ex − e−x ex − e−x e sinh x = 1 = = 2 (ex − e−x ). −1 sinh 1 e −e 2 e − 1/e e −1 12. In this case we have y(0) = c1 = 1, y ′ (1) = 2c2 = 6 so that c1 = 1 and c2 = 3. The solution is y = 1 + 3x2 . 13. From y = c1 ex cos x + c2 ex sin x we find y ′ = c1 ex (− sin x + cos x) + c2 ex (cos x + sin x). (a) We have y(0) = c1 = 1, y ′ (0) = c1 +c2 = 0 so that c1 = 1 and c2 = −1. The solution is y = ex cos x−ex sin x. (b) We have y(0) = c1 = 1, y(π) = −eπ = −1, which is not possible. (c) We have y(0) = c1 = 1, y(π/2) = c2 eπ/2 = 1 so that c1 = 1 and c2 = e−π/2 . The solution is y = ex cos x + e−π/2 ex sin x. (d) We have y(0) = c1 = 0, y(π) = c2 eπ sin π = 0 so that c1 = 0 and c2 is arbitrary. Solutions are y = c2 ex sin x, for any real numbers c2 . 14. (a) We have y(−1) = c1 + c2 + 3 = 0, y(1) = c1 + c2 + 3 = 4, which is not possible. (b) We have y(0) = c1 · 0 + c2 · 0 + 3 = 1, which is not possible. (c) We have y(0) = c1 · 0 + c2 · 0 + 3 = 3, y(1) = c1 + c2 + 3 = 0 so that c1 is arbitrary and c2 = −3 − c1 . Solutions are y = c1 x2 − (c1 + 3)x4 + 3. (d) We have y(1) = c1 + c2 + 3 = 3, y(2) = 4c1 + 16c2 + 3 = 15 so that c1 = −1 and c2 = 1. The solution is y = −x2 + x4 + 3. 15. Since (−4)x + (3)x2 + (1)(4x − 3x2 ) = 0 the set of functions is linearly dependent. 16. Since (1)0 + (0)x + (0)ex = 0 the set of functions is linearly dependent. A similar argument shows that any set of functions containing f (x) = 0 will be linearly dependent. 17. Since (−1/5)5 + (1) cos2 x + (1) sin2 x = 0 the set of functions is linearly dependent. 18. Since (1) cos 2x + (1)1 + (−2) cos2 x = 0 the set of functions is linearly dependent. 19. Since (−4)x + (3)(x − 1) + (1)(x + 3) = 0 the set of functions is linearly dependent. 20. From the graphs of f1 (x) = 2 + x and f2 (x) = 2 + |x| we see that the set of functions is linearly independent since they cannot be multiples of each other. 21. Suppose c1 (1 + x) + c2 x + c3 x2 = 0. Then c1 + (c1 + c2 )x + c3 x2 = 0 and so c1 = 0, c1 + c2 = 0, and c3 = 0. Since c1 = 0 we also have c2 = 0. Thus, the set of functions is linearly independent. 22. Since (−1/2)ex + (1/2)e−x + (1) sinh x = 0 the set of functions is linearly dependent. 100 3.1 Preliminary Theory: Linear Equations 23. The functions satisfy the differential equation and are linearly independent since ! " W e−3x , e4x = 7ex ̸= 0 for −∞ < x < ∞. The general solution is y = c1 e−3x + c2 e4x . 24. The functions satisfy the differential equation and are linearly independent since W (cosh 2x, sinh 2x) = 2 for −∞ < x < ∞. The general solution is y = c1 cosh 2x + c2 sinh 2x. 25. The functions satisfy the differential equation and are linearly independent since W (ex cos 2x, ex sin 2x) = 2e2x ̸= 0 for −∞ < x < ∞. The general solution is y = c1 ex cos 2x + c2 ex sin 2x. 26. The functions satisfy the differential equation and are linearly independent since # $ W ex/2 , xex/2 = ex ̸= 0 for −∞ < x < ∞. The general solution is y = c1 ex/2 + c2 xex/2 . 27. The functions satisfy the differential equation and are linearly independent since ! " W x3 , x4 = x6 ̸= 0 for 0 < x < ∞. The general solution on this interval is y = c1 x3 + c2 x4 . 28. The functions satisfy the differential equation and are linearly independent since W (cos(ln x), sin(ln x)) = 1/x ̸= 0 for 0 < x < ∞. The general solution on this interval is y = c1 cos(ln x) + c2 sin(ln x). 29. The functions satisfy the differential equation and are linearly independent since ! " W x, x−2 , x−2 ln x = 9x−6 ̸= 0 for 0 < x < ∞. The general solution on this interval is y = c1 x + c2 x−2 + c3 x−2 ln x. 30. The functions satisfy the differential equation and are linearly independent since W (1, x, cos x, sin x) = 1 for −∞ < x < ∞. The general solution on this interval is y = c1 + c2 x + c3 cos x + c4 sin x. 101 3.1 Preliminary Theory: Linear Equations 31. The functions y1 = e2x and y2 = e5x form a fundamental set of solutions of the associated homogeneous equation, and yp = 6ex is a particular solution of the nonhomogeneous equation. 32. The functions y1 = cos x and y2 = sin x form a fundamental set of solutions of the associated homogeneous equation, and yp = x sin x + (cos x) ln(cos x) is a particular solution of the nonhomogeneous equation. 33. The functions y1 = e2x and y2 = xe2x form a fundamental set of solutions of the associated homogeneous equation, and yp = x2 e2x + x − 2 is a particular solution of the nonhomogeneous equation. 34. The functions y1 = x−1/2 and y2 = x−1 form a fundamental set of solutions of the associated homogeneous 1 2 equation, and yp = 15 x − 61 x is a particular solution of the nonhomogeneous equation. 35. (a) We have yp′ 1 = 6e2x and yp′′1 = 12e2x , so yp′′1 − 6yp′ 1 + 5yp1 = 12e2x − 36e2x + 15e2x = −9e2x . Also, yp′ 2 = 2x + 3 and yp′′2 = 2, so yp′′2 − 6yp′ 2 + 5yp2 = 2 − 6(2x + 3) + 5(x2 + 3x) = 5x2 + 3x − 16. (b) By the superposition principle for nonhomogeneous equations a particular solution of y ′′ − 6y ′ + 5y = 5x2 + 3x − 16 − 9e2x is yp = x2 + 3x + 3e2x . A particular solution of the second equation is 1 1 yp = −2yp2 − yp1 = −2x2 − 6x − e2x . 9 3 36. (a) yp1 = 5 (b) yp2 = −2x (c) yp = yp1 + yp2 = 5 − 2x (d) yp = 12 yp1 − 2yp2 = 5 2 + 4x 37. (a) Since D2 x = 0, x and 1 are solutions of y ′′ = 0. Since they are linearly independent, the general solution is y = c1 x + c2 . (b) Since D3 x2 = 0, x2 , x, and 1 are solutions of y ′′′ = 0. Since they are linearly independent, the general solution is y = c1 x2 + c2 x + c3 . (c) Since D4 x3 = 0, x3 , x2 , x, and 1 are solutions of y (4) = 0. Since they are linearly independent, the general solution is y = c1 x3 + c2 x2 + c3 x + c4 . (d) By part (a), the general solution of y ′′ = 0 is yc = c1 x + c2 . Since D2 x2 = 2! = 2, yp = x2 is a particular solution of y ′′ = 2. Thus, the general solution is y = c1 x + c2 + x2 . (e) By part (b), the general solution of y ′′′ = 0 is yc = c1 x2 + c2 x + c3 . Since D3 x3 = 3! = 6, yp = x3 is a particular solution of y ′′′ = 6. Thus, the general solution is y = c1 x2 + c2 x + c3 + x3 . (f ) By part (c), the general solution of y (4) = 0 is yc = c1 x3 + c2 x2 + c3 x + c4 . Since D4 x4 = 4! = 24, yp = x4 is a particular solution of y (4) = 24. Thus, the general solution is y = c1 x3 + c2 x2 + c3 x + c4 + x4 . 38. By the superposition principle, if y1 = ex and y2 = e−x are both solutions of a homogeneous linear differential equation, then so are 1 ex + e−x (y1 + y2 ) = = cosh x and 2 2 102 1 ex − e−x (y1 − y2 ) = = sinh x. 2 2 3.2 Reduction of Order 39. (a) From the graphs of y1 = x3 and y2 = |x|3 we see that the functions are linearly independent since they cannot be multiples of each other. It is easily shown that y1 = x3 is a solution of x2 y ′′ − 4xy ′ + 6y = 0. To show that y2 = |x|3 is a solution let y2 = x3 for x ≥ 0 and let y2 = −x3 for x < 0. (b) If x ≥ 0 then y2 = x3 and % 3 % x W (y1 , y2 ) = %% 2 3x If x < 0 then y2 = −x3 and % 3 % x W (y1 , y2 ) = %% 2 3x % x3 % % = 0. 3x2 % % −x3 % % = 0. −3x2 % This does not violate Theorem 3.3 since a2 (x) = x2 is zero at x = 0. (c) The functions Y1 = x3 and Y2 = x2 are solutions of x2 y ′′ − 4xy ′ + 6y = 0. They are linearly independent " ! since W x3 , x2 = x4 ̸= 0 for −∞ < x < ∞. (d) The function y = x3 satisfies y(0) = 0 and y ′ (0) = 0. (e) Neither is the general solution on (−∞, ∞) since we form a general solution on an interval for which a2 (x) ̸= 0 for every x in the interval. 40. Since ex−3 = e−3 ex = (e−5 e2 )ex = e−5 ex+2 , we see that ex−3 is a constant multiple of ex+2 and the set of functions is linearly dependent. 41. Since 0y1 + 0y2 + · · · + 0yk + 1yk+1 = 0, the set of solutions is linearly dependent. 42. The set of solutions is linearly dependent. Suppose n of the solutions are linearly independent (if not, then the set of n + 1 solutions is linearly dependent). Without loss of generality, let this set be y1 , y2 , . . . , yn . Then y = c1 y1 + c2 y2 + · · · + cn yn is the general solution of the nth-order differential equation and for some choice, c∗1 , c∗2 , . . . , c∗n , of the coefficients yn+1 = c∗1 y1 + c∗2 y2 + · · · + c∗n yn . But then the set y1 , y2 , . . . , yn , yn+1 is linearly dependent. EXERCISES 3.2 Reduction of Order In Problems 1-8 we use reduction of order to find a second solution. In Problems 9-16 we use formula (5) from the text. 1. Define y = u(x)e2x so y ′ = 2ue2x + u′ e2x , y ′′ = e2x u′′ + 4e2x u′ + 4e2x u, and y ′′ − 4y ′ + 4y = e2x u′′ = 0. Therefore u′′ = 0 and u = c1 x + c2 . Taking c1 = 1 and c2 = 0 we see that a second solution is y2 = xe2x . 103 3.2 Reduction of Order 2. Define y = u(x)xe−x so y ′ = (1 − x)e−x u + xe−x u′ , y ′′ = xe−x u′′ + 2(1 − x)e−x u′ − (2 − x)e−x u, and y ′′ + 2y ′ + y = e−x (xu′′ + 2u′ ) = 0 or u′′ + 2 ′ u = 0. x 2 If w = u′ we obtain the linear first-order equation w′ + w = 0 which has the integrating factor x & e2 dx/x = x2 . Now d 2 [x w] = 0 gives x2 w = c. dx 1 Therefore w = u′ = c/x2 and u = c1 /x. A second solution is y2 = xe−x = e−x . x 3. Define y = u(x) cos 4x so y ′ = −4u sin 4x + u′ cos 4x, y ′′ = u′′ cos 4x − 8u′ sin 4x − 16u cos 4x and y ′′ + 16y = (cos 4x)u′′ − 8(sin 4x)u′ = 0 or u′′ − 8(tan 4x)u′ = 0. If w& = u′ we obtain the linear first-order equation w′ − 8(tan 4x)w = 0 which has the integrating factor e−8 tan 4x dx = cos2 4x. Now d [(cos2 4x)w] = 0 gives (cos2 4x)w = c. dx Therefore w = u′ = c sec2 4x and u = c1 tan 4x. A second solution is y2 = tan 4x cos 4x = sin 4x. 4. Define y = u(x) sin 3x so y ′ = 3u cos 3x + u′ sin 3x, y ′′ = u′′ sin 3x + 6u′ cos 3x − 9u sin 3x, and y ′′ + 9y = (sin 3x)u′′ + 6(cos 3x)u′ = 0 or u′′ + 6(cot 3x)u′ = 0. If &w = u′ we obtain the linear first-order equation w′ + 6(cot 3x)w = 0 which has the integrating factor e6 cot 3x dx = sin2 3x. Now d [(sin2 3x)w] = 0 gives (sin2 3x)w = c. dx Therefore w = u′ = c csc2 3x and u = c1 cot 3x. A second solution is y2 = cot 3x sin 3x = cos 3x. 5. Define y = u(x) cosh x so y ′ = u sinh x + u′ cosh x, y ′′ = u′′ cosh x + 2u′ sinh x + u cosh x and y ′′ − y = (cosh x)u′′ + 2(sinh x)u′ = 0 or u′′ + 2(tanh x)u′ = 0. If &w = u′ we obtain the linear first-order equation w′ + 2(tanh x)w = 0 which has the integrating factor e2 tanh x dx = cosh2 x. Now d [(cosh2 x)w] = 0 gives (cosh2 x)w = c. dx Therefore w = u′ = c sech2 x and u = c tanh x. A second solution is y2 = tanh x cosh x = sinh x. 6. Define y = u(x)e5x so y ′ = 5e5x u + e5x u′ , y ′′ = e5x u′′ + 10e5x u′ + 25e5x u 104 3.2 Reduction of Order and y ′′ − 25y = e5x (u′′ + 10u′ ) = 0 or u′′ + 10u′ = 0. ′ ′ If w & = u we obtain the linear first-order equation w + 10w = 0 which has the integrating factor 10 dx e = e10x . Now d 10x [e w] = 0 gives e10x w = c. dx Therefore w = u′ = ce−10x and u = c1 e−10x . A second solution is y2 = e−10x e5x = e−5x . 7. Define y = u(x)e2x/3 so y′ = 2 2x/3 e u + e2x/3 u′ , 3 4 4 y ′′ = e2x/3 u′′ + e2x/3 u′ + e2x/3 u 3 9 and 9y ′′ − 12y ′ + 4y = 9e2x/3 u′′ = 0. Therefore u′′ = 0 and u = c1 x + c2 . Taking c1 = 1 and c2 = 0 we see that a second solution is y2 = xe2x/3 . 8. Define y = u(x)ex/3 so y′ = 1 x/3 e u + ex/3 u′ , 3 2 1 y ′′ = ex/3 u′′ + ex/3 u′ + ex/3 u 3 9 and 5 6y ′′ + y ′ − y = ex/3 (6u′′ + 5u′ ) = 0 or u′′ + u′ = 0. 6 5 ′ ′ If w = u we obtain the linear first-order equation w + 6 w = 0 which has the integrating factor & e(5/6) dx = e5x/6 . Now d 5x/6 [e w] = 0 gives e5x/6 w = c. dx Therefore w = u′ = ce−5x/6 and u = c1 e−5x/6 . A second solution is y2 = e−5x/6 ex/3 = e−x/2 . 9. Identifying P (x) = −7/x we have 4 y2 = x ' e− ' e− & (−7/x) dx 4 x8 dx = x ' 1 dx = x4 ln |x|. x A second solution is y2 = x4 ln |x|. 10. Identifying P (x) = 2/x we have 2 y2 = x & (2/x) dx 2 x4 dx = x ' 1 x−6 dx = − x−3 . 5 A second solution is y2 = x−3 . 11. Identifying P (x) = 1/x we have y2 = ln x ' & ( ) ' e− dx/x dx 1 dx = ln x = ln x − = −1. (ln x)2 x(ln x)2 ln x A second solution is y2 = 1. 12. Identifying P (x) = 0 we have y2 = x1/2 ln x ' & ( ) 1 e− 0 dx 1/2 = −x1/2 . dx = x ln x − x(ln x)2 ln x A second solution is y2 = x1/2 . 105 3.2 Reduction of Order 13. Identifying P (x) = −1/x we have y2 = x sin(ln x) = x sin(ln x) ' ' & ' x e− −dx/x dx = x sin(ln x) dx x2 sin2 (ln x) x2 sin2 (ln x) csc2 (ln x) dx = [x sin(ln x)] [− cot(ln x)] = −x cos(ln x). x A second solution is y2 = x cos(ln x). 14. Identifying P (x) = −3/x we have 2 y2 = x cos(ln x) = x2 cos(ln x) ' ' & ' x3 e− −3 dx/x 2 cos(ln x) dx = x dx 4 2 4 x cos (ln x) x cos2 (ln x) sec2 (ln x) dx = x2 cos(ln x) tan(ln x) = x2 sin(ln x). x A second solution is y2 = x2 sin(ln x). ! " 15. Identifying P (x) = 2(1 + x)/ 1 − 2x − x2 we have & ' ln(1−2x−x2 ) ' − 2(1+x)dx/(1−2x−x2 ) e e y2 = (x + 1) dx = (x + 1) dx 2 (x + 1) (x + 1)2 + ' * ' 2 1 − 2x − x2 dx = (x + 1) − 1 dx = (x + 1) (x + 1)2 (x + 1)2 * + 2 = (x + 1) − − x = −2 − x2 − x. x+1 A second solution is y2 = x2 + x + 2. ! " 16. Identifying P (x) = −2x/ 1 − x2 we have ' ' ' & 2 2 y2 = e− −2x dx/(1−x ) dx = e− ln(1−x ) dx = A second solution is y2 = ln |(1 + x)/(1 − x)|. % % 1 %% 1 + x %% 1 . dx = ln % 1 − x2 2 1 − x% 17. Define y = u(x)e−2x so y ′ = −2ue−2x + u′ e−2x , y ′′ = u′′ e−2x − 4u′ e−2x + 4ue−2x and y ′′ − 4y = e−2x u′′ − 4e−2x u′ = 0 or u′′ − 4u′ = 0. If w& = u′ we obtain the linear first-order equation w′ − 4w = 0 which has the integrating factor e−4 dx = e−4x . Now d −4x [e w] = 0 gives e−4x w = c. dx Therefore w = u′ = ce4x and u = c1 e4x . A second solution is y2 = e−2x e4x = e2x . We see by observation that a particular solution is yp = −1/2. The general solution is 1 y = c1 e−2x + c2 e2x − . 2 18. Define y = u(x) · 1 so y ′ = u′ , y ′′ = u′′ and y ′′ + y ′ = u′′ + u′ = 1. 106 3.2 Reduction of Order & If w = u′ we obtain the linear first-order equation w′ + w = 1 which has the integrating factor e d x [e w] = ex dx gives dx = ex . Now ex w = ex + c. Therefore w = u′ = 1 + ce−x and u = x + c1 e−x + c2 . The general solution is y = u = x + c1 e−x + c2 . 19. Define y = u(x)ex so y ′ = uex + u′ ex , y ′′ = u′′ ex + 2u′ ex + uex and y ′′ − 3y ′ + 2y = ex u′′ − ex u′ = 5e3x . If w = u′ we obtain the linear first-order equation w′ − w = 5e2x which has the integrating factor e− Now d −x [e w] = 5ex dx gives & dx = e−x . e−x w = 5ex + c1 . Therefore w = u′ = 5e2x + c1 ex and u = 25 e2x + c1 ex + c2 . The general solution is y = uex = 5 3x e + c1 e2x + c2 ex . 2 20. Define y = u(x)ex so y ′ = uex + u′ ex , y ′′ = u′′ ex + 2u′ ex + uex and y ′′ − 4y ′ + 3y = ex u′′ − ex u′ = x. If w = u′ we obtain the linear first-order equation w′ − 2w = xe−x which has the integrating factor e− e−2x . Now d −2x 1 1 w] = xe−3x gives e−2x w = − xe−3x − e−3x + c1 . [e dx 3 9 Therefore w = u′ = − 31 xe−x − 19 e−x + c1 e2x and u = y = uex = 1 3 & 2dx = xe−x + 49 e−x + c2 e2x + c3 . The general solution is 4 1 x + + c2 e3x + c3 ex . 3 9 21. (a) For m1 constant, let y1 = em1 x . Then y1′ = m1 em1 x and y1′′ = m21 em1 x . Substituting into the differential equation we obtain ay1′′ + by1′ + cy1 = am21 em1 x + bm1 em1 x + cem1 x = em1 x (am21 + bm1 + c) = 0. Thus, y1 = em1 x will be a solution of the differential equation whenever am21 +bm1 +c = 0. Since a quadratic equation always has at least one real or complex root, the differential equation must have a solution of the form y1 = em1 x . (b) Write the differential equation in the form c b y ′′ + y ′ + y = 0, a a 107 3.2 Reduction of Order and let y1 = em1 x be a solution. Then a second solution is given by ' −bx/a e y2 = em1 x dx e2m1 x ' = em1 x e−(b/a+2m1 )x dx 1 em1 x e−(b/a+2m1 )x b/a + 2m1 1 e−(b/a+m1 )x . =− b/a + 2m1 =− (m1 ̸= −b/2a) Thus, when m1 = ̸ −b/2a, a second solution is given by y2 = em2 x where m2 = −b/a − m1 . When m1 = −b/2a a second solution is given by ' m1 x y2 = e dx = xem1 x . (c) The functions 1 ix cos x = (e − e−ix ) 2i 1 sinh x = (ex − e−x ) cosh x = 2 are all expressible in terms of exponential functions. sin x = 1 ix (e + e−ix ) 2 1 x (e + e−x ) 2 22. We have y1′ = 1 and y1′′ = 0, so xy1′′ − xy1′ + y1 = 0 − x + x = 0 and y1 (x) = x is a solution of the differential equation. Letting y = u(x)y1 (x) = xu(x) we get y ′ = xu′ (x) + u(x) and y ′′ = xu′′ (x) + 2u′ (x). Then xy ′′ − xy ′ + y = x2 u′′ + 2xu′ − x2 u′ − xu + xu = x2 u′′ − (x2 − 2x)u′ = 0. If we make the substitution w = u′ , the linear first-order differential equation becomes x2 w′ − (x2 − x)w = 0, which is separable: dw # 1$ w = 1− dx x dw # 1$ dx = 1− w x ln w = x − ln x + c w = c1 Then u′ = c1 ex /x and u = c1 second solution is & ex . x ex dx/x. To integrate ex /x we use the series representation for ex . Thus, a ' ex dx x ) ' ( 1 1 1 = c1 x 1 + x + x2 + x3 + · · · dx x 2! 3! ) ' ( 1 1 1 = c1 x + 1 + x + x2 + · · · dx x 2! 3! ( ) 1 2 1 3 = c1 x ln x + x + x + x + ··· 2(2!) 3(3!) ( ) 1 3 1 4 2 = c1 x ln x + x + x + x + ··· . 2(2!) 3(3!) y2 = xu(x) = c1 x An interval of definition is probably (0, ∞) because of the ln x term. 108 3.3 Homogeneous Linear Equations with Constant Coefficients 23. (a) We have y ′ = y ′′ = ex , so xy ′′ − (x + 10)y ′ + 10y = xex − (x + 10)ex + 10ex = 0, and y = ex is a solution of the differential equation. (b) By (5) in the text a second solution is & x+10 & & ' ' ' − P (x) dx dx x e e (1+10/x)dx e x x y2 = y1 dx = e dx = e dx y12 e2x e2x ' x+ln x10 ' e x = ex dx = e x10 e−x dx e2x = ex (−3,628,800 − 3,628,800x − 1,814,400x2 − 604,800x3 − 151,200x4 − 30,240x5 − 5,040x6 − 720x7 − 90x8 − 10x9 − x10 )e−x = −3,628,800 − 3,628,800x − 1,814,400x2 − 604,800x3 − 151,200x4 − 30,240x5 − 5,040x6 − 720x7 − 90x8 − 10x9 − x10 . 10 , 1 n 1 (c) By Corollary (A) of Theorem 3.2, − y2 = x is a solution. 10! n! n=0 EXERCISES 3.3 Homogeneous Linear Equations with Constant Coefficients 1. From 4m2 + m = 0 we obtain m = 0 and m = −1/4 so that y = c1 + c2 e−x/4 . 2. From m2 − 36 = 0 we obtain m = 6 and m = −6 so that y = c1 e6x + c2 e−6x . 3. From m2 − m − 6 = 0 we obtain m = 3 and m = −2 so that y = c1 e3x + c2 e−2x . 4. From m2 − 3m + 2 = 0 we obtain m = 1 and m = 2 so that y = c1 ex + c2 e2x . 5. From m2 + 8m + 16 = 0 we obtain m = −4 and m = −4 so that y = c1 e−4x + c2 xe−4x . 6. From m2 − 10m + 25 = 0 we obtain m = 5 and m = 5 so that y = c1 e5x + c2 xe5x . 7. From 12m2 − 5m − 2 = 0 we obtain m = −1/4 and m = 2/3 so that y = c1 e−x/4 + c2 e2x/3 . √ √ √ 8. From m2 + 4m − 1 = 0 we obtain m = −2 ± 5 so that y = c1 e(−2+ 5 )x + c2 e(−2− 5 )x . 9. From m2 + 9 = 0 we obtain m = 3i and m = −3i so that y = c1 cos 3x + c2 sin 3x. √ √ √ √ 10. From 3m2 + 1 = 0 we obtain m = i/ 3 and m = −i/ 3 so that y = c1 cos(x/ 3 ) + c2 (sin x/ 3 ). 11. From m2 − 4m + 5 = 0 we obtain m = 2 ± i so that y = e2x (c1 cos x + c2 sin x). 12. From 2m2 + 2m + 1 = 0 we obtain m = −1/2 ± i/2 so that y = e−x/2 [c1 cos(x/2) + c2 sin(x/2)]. 13. From 3m2 + 2m + 1 = 0 we obtain m = −1/3 ± √ 2 i/3 so that √ √ y = e−x/3 [c1 cos( 2x/3) + c2 sin( 2x/3)]. 109 3.3 Homogeneous Linear Equations with Constant Coefficients 14. From 2m2 − 3m + 4 = 0 we obtain m = 3/4 ± √ 23 i/4 so that √ √ y = e3x/4 [c1 cos( 23x/4) + c2 sin( 23x/4)]. 15. From m3 − 4m2 − 5m = 0 we obtain m = 0, m = 5, and m = −1 so that y = c1 + c2 e5x + c3 e−x . √ 16. From m3 − 1 = 0 we obtain m = 1 and m = −1/2 ± 3 i/2 so that √ √ y = c1 ex + e−x/2 [c2 cos( 3x/2) + c3 sin( 3x/2)]. 17. From m3 − 5m2 + 3m + 9 = 0 we obtain m = −1, m = 3, and m = 3 so that y = c1 e−x + c2 e3x + c3 xe3x . 18. From m3 + 3m2 − 4m − 12 = 0 we obtain m = −2, m = 2, and m = −3 so that y = c1 e−2x + c2 e2x + c3 e−3x . 19. From m3 + m2 − 2 = 0 we obtain m = 1 and m = −1 ± i so that u = c1 et + e−t (c2 cos t + c3 sin t). √ 20. From m3 − m2 − 4 = 0 we obtain m = 2 and m = −1/2 ± 7 i/2 so that √ √ x = c1 e2t + e−t/2 [c2 cos( 7t/2) + c3 sin( 7t/2)]. 21. From m3 + 3m2 + 3m + 1 = 0 we obtain m = −1, m = −1, and m = −1 so that y = c1 e−x + c2 xe−x + c3 x2 e−x . 22. From m3 − 6m2 + 12m − 8 = 0 we obtain m = 2, m = 2, and m = 2 so that y = c1 e2x + c2 xe2x + c3 x2 e2x . √ 23. From m4 + m3 + m2 = 0 we obtain m = 0, m = 0, and m = −1/2 ± 3 i/2 so that √ √ y = c1 + c2 x + e−x/2 [c3 cos( 3x/2) + c4 sin( 3x/2)]. 24. From m4 − 2m2 + 1 = 0 we obtain m = 1, m = 1, m = −1, and m = −1 so that y = c1 ex + c2 xex + c3 e−x + c4 xe−x . √ √ 25. From 16m4 + 24m2 + 9 = 0 we obtain m = ± 3 i/2 and m = ± 3 i/2 so that √ √ √ √ y = c1 cos( 3x/2) + c2 sin( 3x/2) + c3 x cos( 3x/2) + c4 x sin( 3x/2). √ 26. From m4 − 7m2 − 18 = 0 we obtain m = 3, m = −3, and m = ± 2 i so that √ √ y = c1 e3x + c2 e−3x + c3 cos 2x + c4 sin 2x. 27. From m5 + 5m4 − 2m3 − 10m2 + m + 5 = 0 we obtain m = −1, m = −1, m = 1, and m = 1, and m = −5 so that u = c1 e−r + c2 re−r + c3 er + c4 rer + c5 e−5r . 28. From 2m5 − 7m4 + 12m3 + 8m2 = 0 we obtain m = 0, m = 0, m = −1/2, and m = 2 ± 2i so that x = c1 + c2 s + c3 e−s/2 + e2s (c4 cos 2s + c5 sin 2s). 110 3.3 Homogeneous Linear Equations with Constant Coefficients 29. From m2 + 16 = 0 we obtain m = ±4i so that y = c1 cos 4x + c2 sin 4x. If y(0) = 2 and y ′ (0) = −2 then c1 = 2, c2 = −1/2, and y = 2 cos 4x − 1 2 sin 4x. 2 30. From m + 1 = 0 we obtain m = ±i so that y = c1 cos θ + c2 sin θ. If y(π/3) = 0 and y ′ (π/3) = 2 then √ 3 1 c1 + c2 = 0 2 2 √ 1 3 − c1 + c2 = 2, 2 2 √ √ so c1 = − 3, c2 = 1, and y = − 3 cos θ + sin θ. 31. From m2 − 4m − 5 = 0 we obtain m = −1 and m = 5, so that y = c1 e−x + c2 e5x . If y(1) = 0 and y ′ (1) = 2, then c1 e−1 + c2 e5 = 0, −c1 e−1 + 5c2 e5 = 2, so c1 = −e/3, c2 = e−5 /3, and y = − 13 e1−x + 31 e5x−5 . 32. From 4m2 − 4m − 3 = 0 we obtain m = −1/2 and m = 3/2 so that y = c1 e−x/2 + c2 e3x/2 . If y(0) = 1 and 3x/2 . y ′ (0) = 5 then c1 + c2 = 1, − 21 c1 + 32 c2 = 5, so c1 = −7/4, c2 = 11/4, and y = − 47 e−x/2 + 11 4 e √ √ √ 2 −x/2 33. From m + m + 2 = 0 we obtain m = −1/2 ± 7 i/2 so that y = e [c1 cos( 7 x/2) + c2 sin( 7 x/2)]. If y(0) = 0 and y ′ (0) = 0 then c1 = 0 and c2 = 0 so that y = 0. 34. From m2 − 2m + 1 = 0 we obtain m = 1 and m = 1 so that y = c1 ex + c2 xex . If y(0) = 5 and y ′ (0) = 10 then c1 = 5, c1 + c2 = 10 so c1 = 5, c2 = 5, and y = 5ex + 5xex . 35. From m3 + 12m2 + 36m = 0 we obtain m = 0, m = −6, and m = −6 so that y = c1 + c2 e−6x + c3 xe−6x . If y(0) = 0, y ′ (0) = 1, and y ′′ (0) = −7 then c1 + c2 = 0, −6c2 + c3 = 1, so c1 = 5/36, c2 = −5/36, c3 = 1/6, and y = 5 36 − 5 −6x 36 e 36c2 − 12c3 = −7, + 61 xe−6x . 36. From m3 + 2m2 − 5m − 6 = 0 we obtain m = −1, m = 2, and m = −3 so that y = c1 e−x + c2 e2x + c3 e−3x . If y(0) = 0, y ′ (0) = 0, and y ′′ (0) = 1 then c1 + c2 + c3 = 0, −c1 + 2c2 − 3c3 = 0, c1 + 4c2 + 9c3 = 1, so c1 = −1/6, c2 = 1/15, c3 = 1/10, and 1 1 1 y = − e−x + e2x + e−3x . 6 15 10 37. From m2 − 10m + 25 = 0 we obtain m = 5 and m = 5 so that y = c1 e5x + c2 xe5x . If y(0) = 1 and y(1) = 0 then c1 = 1, c1 e5 + c2 e5 = 0, so c1 = 1, c2 = −1, and y = e5x − xe5x . 38. From m2 + 4 = 0 we obtain m = ±2i so that y = c1 cos 2x + c2 sin 2x. If y(0) = 0 and y(π) = 0 then c1 = 0 and y = c2 sin 2x. 39. From m2 + 1 = 0 we obtain m = ±i so that y = c1 cos x + c2 sin x and y ′ = −c1 sin x + c2 cos x. From y ′ (0) = c1 (0) + c2 (1) = c2 = 0 and y ′ (π/2) = −c1 (1) = 0 we find c1 = c2 = 0. A solution of the boundary-value problem is y = 0. 40. From m2 − 2m + 2 = 0 we obtain m = 1 ± i so that y = ex (c1 cos x + c2 sin x). If y(0) = 1 and y(π) = 1 then c1 = 1 and y(π) = eπ cos π = −eπ . Since −eπ ̸= 1, the boundary-value problem has no solution. √ √ 41. The auxiliary equation is m2 − 3 = 0 which has roots − 3 and 3 . By (10) the general solution is y = √ √ √ √ √ √ c1 e 3x + c2 e− 3x . By (11) the general solution is y = c1 cosh 3x + c2 sinh 3x. For y = c1 e 3x + c2 e− 3x the 111 3.3 Homogeneous Linear Equations with Constant Coefficients √ √ √ initial conditions imply c1 + c2 = 1, 3c1 − 3c2 = 5. Solving for c1 and c2 we find c1 = 21 (1 + 5 3 ) and c2 = √ √ √ √ √ √ √ 1 y = 12 (1 + 5 3 )e 3x + 21 (1 − 5 3 )e− 3x . For y = c1 cosh 3x + c2 sinh 3x the initial conditions 2 (1 − 5 3 ) so√ √ √ √ √ imply c1 = 1, 3c2 = 5. Solving for c1 and c2 we find c1 = 1 and c2 = 53 3 so y = cosh 3x + 53 3 sinh 3x. 42. The auxiliary equation is m2 − 1 = 0 which has roots −1 and 1. By (10) the general solution is y = c1 ex + c2 e−x . By (11) the general solution is y = c1 cosh x + c2 sinh x. For y = c1 ex + c2 e−x the boundary conditions imply c1 + c2 = 1, c1 e − c2 e−1 = 0. Solving for c1 and c2 we find c1 = 1/(1 + e2 ) and c2 = e2 /(1 + e2 ) so y = ex /(1 + e2 ) + e2 e−x /(1 + e2 ). For y = c1 cosh x + c2 sinh x the boundary conditions imply c1 = 1, c2 = − tanh 1, so y = cosh x − (tanh 1) sinh x. 43. The auxiliary equation should have two positive roots, y = c1 ek1 x + c2 ek2 x . Thus, the differential equation is (f). so that the solution has the form 44. The auxiliary equation should have one positive and one negative root, so that the solution has the form y = c1 ek1 x + c2 e−k2 x . Thus, the differential equation is (a). 45. The auxiliary equation should have a pair of complex roots α ± βi where α < 0, so that the solution has the form eαx (c1 cos βx + c2 sin βx). Thus, the differential equation is (e). 46. The auxiliary equation should have a repeated negative root, so that the solution has the form y = c1 e−x + c2 xe−x . Thus, the differential equation is (c). 47. The differential equation should have the form y ′′ + k 2 y = 0 where k = 1 so that the period of the solution is 2π. Thus, the differential equation is (d). 48. The differential equation should have the form y ′′ + k 2 y = 0 where k = 2 so that the period of the solution is π. Thus, the differential equation is (b). 49. Since (m − 4)(m + 5)2 = m3 + 6m2 − 15m − 100 the differential equation is y ′′′ + 6y ′′ − 15y ′ − 100y = 0. The differential equation is not unique since any constant multiple of the left-hand side of the differential equation would lead to the auxiliary roots. 50. A third root must be m3 = 3 − i and the auxiliary equation is ( ) ( ) 1 1 11 m+ [m − (3 + i)][m − (3 − i)] = m + (m2 − 6x + 10) = m3 − m2 + 7m + 5. 2 2 2 The differential equation is y ′′′ − 11 ′′ y + 7y ′ + 5y = 0. 2 51. From the solution y1 = e−4x cos x we conclude that m1 = −4 + i and m2 = −4 − i are roots of the auxiliary equation. Hence another solution must be y2 = e−4x sin x. Now dividing the polynomial m3 + 6m2 + m − 34 by .. m − (−4 + i) m − (−4 − i) = m2 + 8m + 17 gives m − 2. Therefore m3 = 2 is the third root of the auxiliary equation, and the general solution of the differential equation is y = c1 e−4x cos x + c2 e−4x sin x + c3 e2x . 52. Factoring the difference of two squares we obtain √ √ m4 + 1 = (m2 + 1)2 − 2m2 = (m2 + 1 − 2 m)(m2 + 1 + 2 m) = 0. √ √ Using the quadratic formula on each factor we get m = ± 2/2± 2 i/2. The solution of the differential equation is √ y(x) = e 2 x/2 / √ √ 0 √ 0 √ 2 2 2 2 c1 cos x + c2 sin x + e− 2 x/2 c3 cos x + c4 sin x . 2 2 2 2 / √ 112 3.3 Homogeneous Linear Equations with Constant Coefficients 53. Using the definition of sinh x and the formula for the cosine of the sum of two angles, we have y = sinh x − 2 cos(x + π/6) 1 # # 1 1 π$ π $2 = ex − e−x − 2 (cos x) cos − (sin x) sin 2 2 6 6 /√ 0 1 1 3 1 cos x − sin x = ex − e−x − 2 2 2 2 2 = 1 x 1 −x √ e − e − 3 cos x + sin x. 2 2 This form of the solution can be obtained from the general solution y = c1 ex + c2 e−x + c3 cos x + c4 sin x by √ choosing c1 = 12 , c2 = − 21 , c3 = − 3 , and c4 = 1. 54. The auxiliary equation is m2 + α = 0 and we consider three cases where λ = 0, λ = α2 > 0, and λ = −α2 < 0: Case I When α = 0 the general solution of the differential equation is y = c1 + c2 x. The boundary conditions imply 0 = y(0) = c1 and 0 = y(π/2) = c2 π/2, so that c1 = c2 = 0 and the problem possesses only the trivial solution. Case II When λ = −α2 < 0 the general solution of the differential equation is y = c1 eα x + c2 e−α x , or alternatively, y = c1 cosh α x + c2 sinh α x. Again, y(0) = 0 implies c1 = 0 so y = c2 sinh α x. The second boundary condition implies 0 = y(π/2) = c2 sinh α π/2 or c2 = 0. In this case also, the problem possesses only the trivial solution. When λ = α2 > 0 the general solution of the differential equation is y = c1 cos α x + c2 sin α x. In this case also, y(0) = 0 yields c1 = 0, so that y = c2 sin α x. The second boundary condition implies Case III 0 = c2 sin α π/2. When α π/2 is an integer multiple of π, that is, when α = 2k for k a nonzero integer, the problem will have nontrivial solutions. Thus, for λ = α2 = 4k 2 the boundary-value problem will have nontrivial solutions y = c2 sin 2kx, where k is a nonzero integer. On the other hand, when α is not an even integer, the boundary-value problem will have only the trivial solution. 55. Applying integration by parts twice we have ' 1 eax f (x) dx = eax f (x) − a 1 = eax f (x) − a 1 = eax f (x) − a ' 1 eax f ′ (x) dx a * + ' 1 1 ax ′ 1 ax ′′ e f (x) − e f (x) dx a a a ' 1 ax ′ 1 e f (x) + 2 eax f ′′ (x) dx. a2 a Collecting the integrals we get ) ( ' 1 1 1 ′′ ax e f (x) − 2 f (x) dx = eax f (x) − 2 eax f ′ (x). a a a In order for the technique to work we need to have ) ( ' ' 1 ′′ ax e f (x) − 2 f (x) dx = k eax f (x) dx a or 1 ′′ f (x) = kf (x), a2 where k = ̸ 0. This is the second-order differential equation f (x) − f ′′ (x) + a2 (k − 1)f (x) = 0. 113 3.3 Homogeneous Linear Equations with Constant Coefficients If k < 1, k ̸= 0, the solution of the differential equation is a pair of exponential functions, in which case the original integrand is an exponential function and does not require integration by parts for its evaluation. Similarly, if k = 1, f ′′ (x) = 0 and f (x) has the form f (x) = ax+b. In this case a single application of integration by parts will suffice. Finally, if k > 1, the solution of the differential equation is √ √ f (x) = c1 cos a k − 1 x + c2 sin a k − 1 x, and we see that the technique will work for linear combinations of cos αx and sin αx. √ 56. (a) The auxiliary equation is m2 − 64/L = 0 which has roots ±8/ L . Thus, the general solution of the √ √ differential equation is x = c1 cosh(8t/ L ) + c2 sinh(8t/ L ). √ (b) Setting x(0) = x0 and x′ (0) = 0 we have c1 = x0 , 8c2 / L = 0. Solving for c1 and c2 we get c1 = x0 and √ c2 = 0, so x(t) = x0 cosh(8t/ L ). √ (c) When L = 20 and x0 = 1, x(t) = cosh(4t/ 5 ). The chain will last touch the peg when x(t) = 10. √ Solving x(t) = 10 for t we get t1 = 41 5 cosh−1 10 ≈ 1.67326. The velocity of the chain at this instant is 3 x′ (t1 ) = 12 11/5 ≈ 17.7989 ft/s. 57. Using a CAS to solve the auxiliary equation m3 − 6m2 + 2m + 1 we find m1 = −0.270534, m2 = 0.658675, and m3 = 5.61186. The general solution is y = c1 e−0.270534x + c2 e0.658675x + c3 e5.61186x . 58. Using a CAS to solve the auxiliary equation 6.11m3 + 8.59m2 + 7.93m + 0.778 = 0 we find m1 = −0.110241, m2 = −0.647826 + 0.857532i, and m3 = −0.647826 − 0.857532i. The general solution is y = c1 e−0.110241x + e−0.647826x (c2 cos 0.857532x + c3 sin 0.857532x). 59. Using a CAS to solve the auxiliary equation 3.15m4 − 5.34m2 + 6.33m − 2.03 = 0 we find m1 = −1.74806, m2 = 0.501219, m3 = 0.62342 + 0.588965i, and m4 = 0.62342 − 0.588965i. The general solution is y = c1 e−1.74806x + c2 e0.501219x + e0.62342x (c3 cos 0.588965x + c4 sin 0.588965x). √ √ 60. Using a CAS to solve the auxiliary equation m4 +2m2 −m+2 = 0 we find m1 = 1/2+ 3 i/2, m2 = 1/2− 3 i/2, √ √ m3 = −1/2 + 7 i/2, and m4 = −1/2 − 7 i/2. The general solution is / / √ √ √ 0 √ 0 3 3 7 7 x/2 −x/2 c1 cos y=e c3 cos x + c2 sin x +e x + c4 sin x . 2 2 2 2 61. From 2m4 + 3m3 − 16m2 + 15m − 4 = 0 we obtain m = −4, m = −4x y = c1 e x/2 + c2 e x x ′ 1 2 ′′ , m = 1, and m = 1, so that + c3 e + c4 xe . If y(0) = −2, y (0) = 6, y (0) = 3, and y ′′′ (0) = c1 + c2 + c3 = −2 1 −4c1 + c2 + c3 + c4 = 6 2 1 16c1 + c2 + c3 + 2c4 = 3 4 1 1 −64c1 + c2 + c3 + 3c4 = , 8 2 4 so c1 = − 75 , c2 = − 116 3 , c3 = 918 25 , c4 = − 58 5 , and y=− 4 −4x 116 x/2 918 x 58 x e − e + e − xe . 75 3 25 5 114 1 2 , then 3.4 Undetermined Coefficients 62. From m4 −3m3 +3m2 −m = 0 we obtain m = 0, m = 1, m = 1, and m = 1 so that y = c1 +c2 ex +c3 xex +c4 x2 ex . If y(0) = 0, y ′ (0) = 0, y ′′ (0) = 1, and y ′′′ (0) = 1 then c1 + c2 = 0, c2 + c3 = 0, c2 + 2c3 + 2c4 = 1, c2 + 3c3 + 6c4 = 1, so c1 = 2, c2 = −2, c3 = 2, c4 = −1/2, and 1 y = 2 − 2ex + 2xex − x2 ex . 2 EXERCISES 3.4 Undetermined Coefficients 1. From m2 + 3m + 2 = 0 we find m1 = −1 and m2 = −2. Then yc = c1 e−x + c2 e−2x and we assume yp = A. Substituting into the differential equation we obtain 2A = 6. Then A = 3, yp = 3 and y = c1 e−x + c2 e−2x + 3. 2. From 4m2 + 9 = 0 we find m1 = − 23 i and m2 = 23 i. Then yc = c1 cos 32 x + c2 sin 23 x and we assume yp = A. Substituting into the differential equation we obtain 9A = 15. Then A = 53 , yp = 53 and 3 5 3 y = c1 cos x + c2 sin x + . 2 2 3 3. From m2 − 10m + 25 = 0 we find m1 = m2 = 5. Then yc = c1 e5x + c2 xe5x and we assume yp = Ax + B. Substituting into the differential equation we obtain 25A = 30 and −10A + 25B = 3. Then A = 56 , B = 65 , yp = 56 x + 56 , and 6 6 y = c1 e5x + c2 xe5x + x + . 5 5 4. From m2 + m − 6 = 0 we find m1 = −3 and m2 = 2. Then yc = c1 e−3x + c2 e2x and we assume yp = Ax + B. 1 , Substituting into the differential equation we obtain −6A = 2 and A − 6B = 0. Then A = − 31 , B = − 18 1 1 yp = − 3 x − 18 , and 1 1 y = c1 e−3x + c2 e2x − x − . 3 18 5. From 14 m2 + m + 1 = 0 we find m1 = m2 = −2. Then yc = c1 e−2x + c2 xe−2x and we assume yp = Ax2 + Bx + C. Substituting into the differential equation we obtain A = 1, 2A + B = −2, and 21 A + B + C = 0. Then A = 1, B = −4, C = 27 , yp = x2 − 4x + 72 , and y = c1 e−2x + c2 xe−2x + x2 − 4x + 7 . 2 6. From m2 − 8m + 20 = 0 we find m1 = 4 + 2i and m2 = 4 − 2i. Then yc = e4x (c1 cos 2x + c2 sin 2x) and we assume yp = Ax2 + Bx + C + (Dx + E)ex . Substituting into the differential equation we obtain 2A − 8B + 20C = 0 −6D + 13E = 0 −16A + 20B = 0 13D = −26 20A = 100. 115 3.4 Undetermined Coefficients " x ! + −2x − 12 13 e and ) ( 12 x 11 4x 2 e . + −2x − y = e (c1 cos 2x + c2 sin 2x) + 5x + 4x + 10 13 √ √ √ √ 7. From m2 + 3 = 0 we find m1 = 3 i and m2 = − 3 i. Then yc = c1 cos 3 x + c2 sin 3 x and we assume Then A = 5, B = 4, C = 11 10 2 , D = −2, E = − 12 13 , yp = 5x + 4x + 11 10 yp = (Ax2 +Bx+C)e3x . Substituting into the differential equation we obtain 2A+6B +12C = 0, 12A+12B = 0, " ! and 12A = −48. Then A = −4, B = 4, C = − 34 , yp = −4x2 + 4x − 34 e3x and ( ) √ √ 4 3x 2 y = c1 cos 3 x + c2 sin 3 x + −4x + 4x − e . 3 8. From 4m2 − 4m − 3 = 0 we find m1 = 23 and m2 = − 21 . Then yc = c1 e3x/2 + c2 e−x/2 and we assume yp = A cos 2x + B sin 2x. Substituting into the differential equation we obtain −19 − 8B = 1 and 8A − 19B = 0. 8 19 19 , B = − 425 , yp = − 425 cos 2x − Then A = − 425 8 425 sin 2x, and y = c1 e3x/2 + c2 e−x/2 − 19 8 cos 2x − sin 2x. 425 425 9. From m2 − m = 0 we find m1 = 1 and m2 = 0. Then yc = c1 ex + c2 and we assume yp = Ax. Substituting into the differential equation we obtain −A = −3. Then A = 3, yp = 3x and y = c1 ex + c2 + 3x. 10. From m2 +2m = 0 we find m1 = −2 and m2 = 0. Then yc = c1 e−2x +c2 and we assume yp = Ax2 +Bx+Cxe−2x . Substituting into the differential equation we obtain 2A + 2B = 5, 4A = 2, and −2C = −1. Then A = 21 , B = 2, C = 1 2 , yp = 12 x2 + 2x + 12 xe−2x , and 1 1 y = c1 e−2x + c2 + x2 + 2x + xe−2x . 2 2 11. From m2 − m + 41 = 0 we find m1 = m2 = 12 . Then yc = c1 ex/2 + c2 xex/2 and we assume yp = A + Bx2 ex/2 . Substituting into the differential equation we obtain 41 A = 3 and 2B = 1. Then A = 12, B = 21 , yp = 12 + 12 x2 ex/2 , and 1 y = c1 ex/2 + c2 xex/2 + 12 + x2 ex/2 . 2 12. From m2 − 16 = 0 we find m1 = 4 and m2 = −4. Then yc = c1 e4x + c2 e−4x and we assume yp = Axe4x . Substituting into the differential equation we obtain 8A = 2. Then A = 41 , yp = 14 xe4x and 1 y = c1 e4x + c2 e−4x + xe4x . 4 13. From m2 + 4 = 0 we find m1 = 2i and m2 = −2i. Then yc = c1 cos 2x + c2 sin 2x and we assume yp = Ax cos 2x + Bx sin 2x. Substituting into the differential equation we obtain 4B = 0 and −4A = 3. Then A = − 34 , B = 0, yp = − 43 x cos 2x, and 3 y = c1 cos 2x + c2 sin 2x − x cos 2x. 4 14. From m2 − 4 = 0 we find m1 = 2 and m2 = −2. Then yc = c1 e2x + c2 e−2x and we assume that yp = (Ax2 + Bx + C) cos 2x + (Dx2 + Ex + F ) sin 2x. Substituting into the differential equation we obtain −8A = 0 −8B + 8D = 0 2A − 8C + 4E = 0 −8D = 1 −8A − 8E = 0 −4B + 2D − 8F = −3. 116 3.4 Undetermined Coefficients ! , so yp = − 18 x cos 2x + − 81 x2 + ( ) 1 1 2 13 2x −2x y = c1 e + c2 e − x cos 2x + − x + sin 2x. 8 8 32 Then A = 0, B = − 18 , C = 0, D = − 18 , E = 0, F = 13 32 15. From m2 + 1 = 0 we find m1 = i and m2 = −i. 2 13 32 " sin 2x, and Then yc = c1 cos x + c2 sin x and we assume 2 yp = (Ax + Bx) cos x + (Cx + Dx) sin x. Substituting into the differential equation we obtain 4C = 0, 2A + 2D = 0, −4A = 2, and −2B + 2C = 0. Then A = − 21 , B = 0, C = 0, D = 12 , yp = − 12 x2 cos x + 21 x sin x, and 1 1 y = c1 cos x + c2 sin x − x2 cos x + x sin x. 2 2 16. From m2 −5m = 0 we find m1 = 5 and m2 = 0. Then yc = c1 e5x +c2 and we assume yp = Ax4 +Bx3 +Cx2 +Dx. Substituting into the differential equation we obtain −20A = 2, 12A − 15B = −4, 6B − 10C = −1, and 14 53 1 4 14 3 53 2 697 1 , B = 75 , C = 250 , D = − 697 2C − 5D = 6. Then A = − 10 625 , yp = − 10 x + 75 x + 250 x − 625 x, and y = c1 e5x + c2 − 1 4 14 3 53 2 697 x + x + x − x. 10 75 250 625 17. From m2 − 2m + 5 = 0 we find m1 = 1 + 2i and m2 = 1 − 2i. Then yc = ex (c1 cos 2x + c2 sin 2x) and we assume yp = Axex cos 2x + Bxex sin 2x. Substituting into the differential equation we obtain 4B = 1 and −4A = 0. Then A = 0, B = 14 , yp = 14 xex sin 2x, and 1 y = ex (c1 cos 2x + c2 sin 2x) + xex sin 2x. 4 18. From m2 − 2m + 2 = 0 we find m1 = 1 + i and m2 = 1 − i. Then yc = ex (c1 cos x + c2 sin x) and we assume yp = Ae2x cos x+Be2x sin x. Substituting into the differential equation we obtain A+2B = 1 and −2A+B = −3. Then A = 75 , B = − 15 , yp = 57 e2x cos x − 51 e2x sin x and 7 1 y = ex (c1 cos x + c2 sin x) + e2x cos x − e2x sin x. 5 5 19. From m2 +2m+1 = 0 we find m1 = m2 = −1. Then yc = c1 e−x +c2 xe−x and we assume yp = A cos x+B sin x+ C cos 2x + D sin 2x. Substituting into the differential equation we obtain 2B = 0, −2A = 1, −3C + 4D = 3, and 9 12 9 12 −4C − 3D = 0. Then A = − 21 , B = 0, C = − 25 , D = 25 , yp = − 12 cos x − 25 cos 2x + 25 sin 2x, and y = c1 e−x + c2 xe−x − 1 9 12 cos x − cos 2x + sin 2x. 2 25 25 20. From m2 + 2m − 24 = 0 we find m1 = −6 and m2 = 4. Then yc = c1 e−6x + c2 e4x and we assume yp = A + (Bx2 + Cx)e4x . Substituting into the differential equation we obtain −24A = 16, 2B + 10C = −2, !1 2 " 1 19 19 and 20B = −1. Then A = − 32 , B = − 20 , C = − 100 , yp = − 23 − 20 x + 100 x e4x , and ( ) 1 2 19 2 x + x e4x . y = c1 e−6x + c2 e4x − − 3 20 100 21. From m3 − 6m2 = 0 we find m1 = m2 = 0 and m3 = 6. Then yc = c1 + c2 x + c3 e6x and we assume yp = Ax2 + B cos x + C sin x. Substituting into the differential equation we obtain −12A = 3, 6B − C = −1, 6 ,C= and B + 6C = 0. Then A = − 14 , B = − 37 1 37 , yp = − 41 x2 − 6 37 cos x + 1 37 6 1 1 cos x + sin x. y = c1 + c2 x + c3 e6x − x2 − 4 37 37 117 sin x, and 3.4 Undetermined Coefficients 22. From m3 − 2m2 − 4m + 8 = 0 we find m1 = m2 = 2 and m3 = −2. Then yc = c1 e2x + c2 xe2x + c3 e−2x and we assume yp = (Ax3 + Bx2 )e2x . Substituting into the differential equation we obtain 24A = 6 and 6A + 8B = 0. " ! 3 3 2 Then A = 41 , B = − 16 , yp = 41 x3 − 16 x e2x , and ) ( 3 1 3 y = c1 e2x + c2 xe2x + c3 e−2x + x − x2 e2x . 4 16 23. From m3 − 3m2 + 3m − 1 = 0 we find m1 = m2 = m3 = 1. Then yc = c1 ex + c2 xex + c3 x2 ex and we assume yp = Ax + B + Cx3 ex . Substituting into the differential equation we obtain −A = 1, 3A − B = 0, and 6C = −4. Then A = −1, B = −3, C = − 32 , yp = −x − 3 − 32 x3 ex , and 2 y = c1 ex + c2 xex + c3 x2 ex − x − 3 − x3 ex . 3 24. From m3 − m2 − 4m + 4 = 0 we find m1 = 1, m2 = 2, and m3 = −2. Then yc = c1 ex + c2 e2x + c3 e−2x and we assume yp = A + Bxex + Cxe2x . Substituting into the differential equation we obtain 4A = 5, −3B = −1, and 4C = 1. Then A = 5 4 ,B= 1 3 ,C= 1 4 , yp = 5 4 + 31 xex + 14 xe2x , and y = c1 ex + c2 e2x + c3 e−2x + 5 1 x 1 2x + xe + xe . 4 3 4 25. From m4 +2m2 +1 = 0 we find m1 = m3 = i and m2 = m4 = −i. Then yc = c1 cos x+c2 sin x+c3 x cos x+c4 x sin x and we assume yp = Ax2 + Bx + C. Substituting into the differential equation we obtain A = 1, B = −2, and 4A + C = 1. Then A = 1, B = −2, C = −3, yp = x2 − 2x − 3, and y = c1 cos x + c2 sin x + c3 x cos x + c4 x sin x + x2 − 2x − 3. 26. From m4 − m2 = 0 we find m1 = m2 = 0, m3 = 1, and m4 = −1. Then yc = c1 + c2 x + c3 ex + c4 e−x and we assume yp = Ax3 + Bx2 + (Cx2 + Dx)e−x . Substituting into the differential equation we obtain −6A = 4, ! " −2B = 0, 10C −2D = 0, and −4C = 2. Then A = − 32 , B = 0, C = − 21 , D = − 52 , yp = − 32 x3 − 21 x2 + 52 x e−x , and x −x y = c1 + c2 x + c3 e + c4 e 2 − x3 − 3 ( ) 1 2 5 x + x e−x . 2 2 27. We have yc = c1 cos 2x + c2 sin 2x and we assume yp = A. Substituting into the differential equation we find √ A = − 12 . Thus y = c1 cos 2x + c2 sin 2x − 12 . From the initial conditions we obtain c1 = 0 and c2 = 2 , so y= √ 2 sin 2x − 1 . 2 28. We have yc = c1 e−2x + c2 ex/2 and we assume yp = Ax2 + Bx + C. Substituting into the differential equation we find A = −7, B = −19, and C = −37. Thus y = c1 e−2x + c2 ex/2 − 7x2 − 19x − 37. From the initial conditions we obtain c1 = − 15 and c2 = 186 5 , so 186 x/2 1 e − 7x2 − 19x − 37. y = − e−2x + 5 5 29. We have yc = c1 e−x/5 + c2 and we assume yp = Ax2 + Bx. Substituting into the differential equation we find A = −3 and B = 30. Thus y = c1 e−x/5 + c2 − 3x2 + 30x. From the initial conditions we obtain c1 = 200 and c2 = −200, so y = 200e−x/5 − 200 − 3x2 + 30x. 118 3.4 Undetermined Coefficients 30. We have yc = c1 e−2x +c2 xe−2x and we assume yp = (Ax3 +Bx2 )e−2x . Substituting into the differential equation " ! we find A = 16 and B = 23 . Thus y = c1 e−2x + c2 xe−2x + 16 x3 + 23 x2 e−2x . From the initial conditions we obtain c1 = 2 and c2 = 9, so ) ( 1 3 3 2 −2x x + x e . y = 2e−2x + 9xe−2x + 6 2 31. We have yc = e−2x (c1 cos x + c2 sin x) and we assume yp = Ae−4x . Substituting into the differential equation we find A = 7. Thus y = e−2x (c1 cos x + c2 sin x) + 7e−4x . From the initial conditions we obtain c1 = −10 and c2 = 9, so y = e−2x (−10 cos x + 9 sin x) + 7e−4x . 32. We have yc = c1 cosh x + c2 sinh x and we assume yp = Ax cosh x + Bx sinh x. Substituting into the differential equation we find A = 0 and B = 12 . Thus 1 y = c1 cosh x + c2 sinh x + x sinh x. 2 From the initial conditions we obtain c1 = 2 and c2 = 12, so 1 y = 2 cosh x + 12 sinh x + x sinh x. 2 33. We have xc = c1 cos ωt + c2 sin ωt and we assume xp = At cos ωt + Bt sin ωt. Substituting into the differential equation we find A = −F0 /2ω and B = 0. Thus x = c1 cos ωt + c2 sin ωt − (F0 /2ω)t cos ωt. From the initial conditions we obtain c1 = 0 and c2 = F0 /2ω 2 , so x = (F0 /2ω 2 ) sin ωt − (F0 /2ω)t cos ωt. 34. We have xc = c1 cos ωt + c2 sin ωt and we assume xp = A cos γt + B sin γt, where γ ̸= ω. Substituting into the differential equation we find A = F0 /(ω 2 − γ 2 ) and B = 0. Thus x = c1 cos ωt + c2 sin ωt + F0 cos γt. ω2 − γ 2 From the initial conditions we obtain c1 = −F0 /(ω 2 − γ 2 ) and c2 = 0, so x=− F0 F0 cos ωt + 2 cos γt. ω2 − γ 2 ω − γ2 35. We have yc = c1 + c2 ex + c3 xex and we assume yp = Ax + Bx2 ex + Ce5x . Substituting into the differential equation we find A = 2, B = −12, and C = 21 . Thus 1 y = c1 + c2 ex + c3 xex + 2x − 12x2 ex + e5x . 2 From the initial conditions we obtain c1 = 11, c2 = −11, and c3 = 9, so 1 y = 11 − 11ex + 9xex + 2x − 12x2 ex + e5x . 2 √ √ 36. We have yc = c1 e−2x + ex (c2 cos 3 x + c3 sin 3 x) and we assume yp = Ax + B + Cxe−2x . Substituting into the differential equation we find A = 41 , B = − 58 , and C = 32 . Thus √ √ 5 2 1 3 x) + x − + xe−2x . 4 8 3 √ 23 59 17 From the initial conditions we obtain c1 = − 12 , c2 = − 24 , and c3 = 72 3 , so ) ( √ √ 23 17 √ 1 5 2 59 y = − e−2x + ex − cos 3 x + 3 sin 3 x + x − + xe−2x . 12 24 72 4 8 3 y = c1 e−2x + ex (c2 cos 3 x + c3 sin 119 3.4 Undetermined Coefficients 37. We have yc = c1 cos x + c2 sin x and we assume yp = A2 + Bx + C. Substituting into the differential equation we find A = 1, B = 0, and C = −1. Thus y = c1 cos x + c2 sin x + x2 − 1. From y(0) = 5 and y(1) = 0 we obtain c1 − 1 = 5 (cos 1)c1 + (sin 1)c2 = 0. Solving this system we find c1 = 6 and c2 = −6 cot 1. The solution of the boundary-value problem is y = 6 cos x − 6(cot 1) sin x + x2 − 1. 38. We have yc = ex (c1 cos x + c2 sin x) and we assume yp = Ax + B. Substituting into the differential equation we find A = 1 and B = 0. Thus y = ex (c1 cos x + c2 sin x) + x. From y(0) = 0 and y(π) = π we obtain c1 = 0 π π − e c1 = π. Solving this system we find c1 = 0 and c2 is any real number. The solution of the boundary-value problem is y = c2 ex sin x + x. √ √ 39. The general solution of the differential equation y ′′ + 3y = 6x is y = c1 cos 3x + c2 sin 3x + 2x. The √ condition y(0) = 0 implies c1 = 0 and so y = c2 sin 3x + 2x. The condition y(1) + y ′ (1) = 0 implies √ √ √ √ √ √ c2 sin 3 + 2 + c2 3 cos 3 + 2 = 0 so c2 = −4/(sin 3 + 3 cos 3 ). The solution is √ −4 sin 3x √ √ √ + 2x. y= sin 3 + 3 cos 3 √ √ 40. Using the general solution y = c1 cos 3x + c2 sin 3x + 2x, the boundary conditions y(0) + y ′ (0) = 0, y(1) = 0 yield the system √ c1 + 3c2 + 2 = 0 √ √ c1 cos 3 + c2 sin 3 + 2 = 0. Solving gives Thus, √ √ 2(− 3 + sin 3 ) √ √ c1 = √ 3 cos 3 − sin 3 and c2 = √ √ 2(1 − cos 3 ) √ √ . 3 cos 3 − sin 3 √ √ √ √ √ 2(− 3 + sin 3 ) cos 3x 2(1 − cos 3 ) sin 3x √ √ √ √ + 2x. √ y= + √ 3 cos 3 − sin 3 3 cos 3 − sin 3 41. We have yc = c1 cos 2x + c2 sin 2x and we assume yp = A cos x + B sin x on [0, π/2]. Substituting into the differential equation we find A = 0 and B = 1 3 . Thus y = c1 cos 2x + c2 sin 2x + 31 sin x on [0, π/2]. On (π/2, ∞) we have y = c3 cos 2x + c4 sin 2x. From y(0) = 1 and y ′ (0) = 2 we obtain c1 = 1 1 + 2c2 = 2. 3 Solving this system we find c1 = 1 and c2 = 56 . Thus y = cos 2x + 65 sin 2x + 13 sin x on [0, π/2]. Now continuity of y at x = π/2 implies 5 1 π cos π + sin π + sin = c3 cos π + c4 sin π 6 3 2 2 1 ′ or −1 + 3 = −c3 . Hence c3 = 3 . Continuity of y at x = π/2 implies −2 sin π + 5 1 π cos π + cos = −2c3 sin π + 2c4 cos π 3 3 2 120 3.4 or − 53 = −2c4 . Then c4 = 5 6 Undetermined Coefficients and the solution of the initial-value problem is 4 cos 2x + 65 sin 2x + 31 sin x, 0 ≤ x ≤ π/2 y(x) = 5 2 x > π/2. 3 cos 2x + 6 sin 2x, 42. We have yc = ex (c1 cos 3x + c2 sin 3x) and we assume yp = A on [0, π]. Substituting into the differential equation we find A = 2. Thus, y = ex (c1 cos 3x + c2 sin 3x) + 2 on [0, π]. On (π, ∞) we have y = ex (c3 cos 3x + c4 sin 3x). From y(0) = 0 and y ′ (0) = 0 we obtain c1 = −2, Solving this system, we find c1 = −2 and c2 = 2 3 c1 + 3c2 = 0. . Thus y = ex (−2 cos 3x + 2 3 sin 3x) + 2 on [0, π]. Now, continuity of y at x = π implies eπ (−2 cos 3π + 2 sin 3π) + 2 = eπ (c3 cos 3π + c4 sin 3π) 3 or 2 + 2eπ = −c3 eπ or c3 = −2e−π (1 + eπ ). Continuity of y ′ at π implies 20 π e sin 3π = eπ [(c3 + 3c4 ) cos 3π + (−3c3 + c4 ) sin 3π] 3 or −c3 eπ − 3c4 eπ = 0. Since c3 = −2e−π (1 + eπ ) we have c4 = 23 e−π (1 + eπ ). The solution of the initial-value problem is 4 x e (−2 cos 3x + 32 sin 3x) + 2, 0≤x≤π y(x) = 2 π x−π (1 + e )e (−2 cos 3x + 3 sin 3x), x > π. 43. (a) From yp = Aekx we find yp′ = Akekx and yp′′ = Ak 2 ekx . Substituting into the differential equation we get aAk 2 ekx + bAkekx + cAekx = (ak 2 + bk + c)Aekx = ekx , so (ak 2 + bk + c)A = 1. Since k is not a root of am2 + bm + c = 0, A = 1/(ak 2 + bk + c). (b) From yp = Axekx we find yp′ = Akxekx +Aekx and yp′′ = Ak 2 xekx +2Akekx . Substituting into the differential equation we get aAk 2 xekx + 2aAkekx + bAkxekx + bAekx + cAxekx = (ak 2 + bk + c)Axekx + (2ak + b)Aekx = (0)Axekx + (2ak + b)Aekx = (2ak + b)Aekx = ekx where ak 2 +bk +c = 0 because k is a root of the auxiliary equation. Now, the roots of the auxiliary equation √ are −b/2a ± b2 − 4ac /2a, and since k is a root of multiplicity one, k ̸= −b/2a and 2ak + b ̸= 0. Thus (2ak + b)A = 1 and A = 1/(2ak + b). (c) If k is a root of multiplicity two, then, as we saw in part (b), k = −b/2a and 2ak + b = 0. From yp = Ax2 ekx we find yp′ = Akx2 ekx + 2Axekx and yp′′ = Ak 2 x2 ekx + 4Akxekx = 2Aekx . Substituting into the differential equation, we get aAk 2 x2 ekx + 4aAkxekx + 2aAekx + bAkx2 ekx + 2bAxekx + cAx2 ekx = (ak 2 + bk + c)Ax2 ekx + 2(2ak + b)Axekx + 2aAekx = (0)Ax2 ekx + 2(0)Axekx + 2aAekx = 2aAekx = ekx . Since the differential equation is second order, a ̸= 0 and A = 1/(2a). 44. Using the double-angle formula for the cosine, we have sin x cos 2x = sin x(cos2 x − sin2 x) = sin x(1 − 2 sin2 x) = sin x − 2 sin3 x. 121 3.4 Undetermined Coefficients Since sin x is a solution of the related homogeneous differential equation we look for a particular solution of the form yp = Ax sin x + Bx cos x + C sin3 x. Substituting into the differential equation we obtain 2A cos x + (6C − 2B) sin x − 8C sin3 x = sin x − 2 sin3 x. Equating coefficients we find A = 0, C = 1 4 , and B = yp = 1 4 . Thus, a particular solution is 1 1 x cos x + sin3 x. 4 4 45. (a) f (x) = ex sin x. We see that yp → ∞ as x → ∞ and yp → 0 as x → −∞. (b) f (x) = e−x . We see that yp → ∞ as x → ∞ and yp → ∞ as x → −∞. (c) f (x) = sin 2x. We see that yp is sinusoidal. (d) f (x) = 1. We see that yp is constant and simply translates yc vertically. 46. The complementary function is yc = e2x (c1 cos 2x + c2 sin 2x). We assume a particular solution of the form yp = (Ax3 + Bx2 + Cx)e2x cos 2x + (Dx3 + Ex2 + F )e2x sin 2x. Substituting into the differential equation and using a CAS to simplify yields [12Dx2 + (6A + 8E)x + (2B + 4F )]e2x cos 2x + [−12Ax2 + (−8B + 6D)x + (−4C + 2E)]e2x sin 2x = (2x2 − 3x)e2x cos 2x + (10x2 − x − 1)e2x sin 2x. This gives the system of equations 12D = 2, −12A = 10, 6A + 8E = −3, −8B + 6D = −1, 2B + 4F = 0, −4C + 2E = −1, from which we find A = − 65 , B = 14 , C = 38 , D = 16 , E = 41 , and F = − 18 . Thus, a particular solution of the differential equation is ( ( ) ) 5 1 3 1 3 1 2 1 yp = − x3 + x2 + x e2x cos 2x + x + x − x e2x sin 2x. 6 4 8 6 4 8 47. The complementary function is yc = c1 cos x + c2 sin x + c3 x cos x + c4 x sin x. We assume a particular solution of the form yp = Ax2 cos x + Bx3 sin x. Substituting into the differential equation and using a CAS to simplify yields (−8A + 24B) cos x + 3Bx sin x = 2 cos x − 3x sin x. This implies −8A + 24B = 2 and −24B = −3. Thus B = 122 1 8 ,A= 1 8 , and yp = 18 x2 cos x + 81 x3 sin x. 3.5 Variation of Parameters EXERCISES 3.5 Variation of Parameters The particular solution, yp = u1 y1 + u2 y2 , in the following problems can take on a variety of forms, especially where trigonometric functions are involved. The validity of a particular form can best be checked by substituting it back into the differential equation. 1. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % % = 1. W =% − sin x cos x % Identifying f (x) = sec x we obtain sin x sec x = − tan x 1 cos x sec x = 1. u′2 = 1 u′1 = − Then u1 = ln | cos x|, u2 = x, and y = c1 cos x + c2 sin x + cos x ln | cos x| + x sin x. 2. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % = 1. % W =% − sin x cos x % Identifying f (x) = tan x we obtain u′1 = − sin x tan x = cos2 x − 1 = cos x − sec x cos x u′2 = sin x. Then u1 = sin x − ln | sec x + tan x|, u2 = − cos x, and y = c1 cos x + c2 sin x + cos x (sin x − ln | sec x + tan x|) − cos x sin x = c1 cos x + c2 sin x − cos x ln | sec x + tan x|. 3. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % % = 1. W =% − sin x cos x % Identifying f (x) = sin x we obtain u′1 = − sin2 x u′2 = cos x sin x. Then u1 = 1 1 1 1 sin 2x − x = sin x cos x − x 4 2 2 2 1 u2 = − cos2 x. 2 123 3.5 Variation of Parameters and 1 1 1 sin x cos2 x − x cos x − cos2 x sin x 2 2 2 1 = c1 cos x + c2 sin x − x cos x. 2 y = c1 cos x + c2 sin x + 4. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % % = 1. W =% − sin x cos x % Identifying f (x) = sec x tan x we obtain u′1 = − sin x(sec x tan x) = − tan2 x = 1 − sec2 x u′2 = cos x(sec x tan x) = tan x. Then u1 = x − tan x, u2 = − ln | cos x|, and y = c1 cos x + c2 sin x + x cos x − sin x − sin x ln | cos x| = c1 cos x + c3 sin x + x cos x − sin x ln | cos x|. 5. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % % = 1. W =% − sin x cos x % Identifying f (x) = cos2 x we obtain u′1 = − sin x cos2 x Then u1 = 1 3 cos3 x, u2 = sin x − 1 3 ! " u′2 = cos3 x = cos x 1 − sin2 x . sin3 x, and 1 1 cos4 x + sin2 x − sin4 x 3 3 " ! " 1! 2 = c1 cos x + c2 sin x + cos x + sin2 x cos2 x − sin2 x + sin2 x 3 2 1 = c1 cos x + c2 sin x + cos2 x + sin2 x 3 3 1 1 = c1 cos x + c2 sin x + + sin2 x. 3 3 y = c1 cos x + c2 sin x + 6. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % = 1. W = %% − sin x cos x % Identifying f (x) = sec2 x we obtain u′1 = − sin x cos2 x u′2 = sec x. Then u1 = − 1 = − sec x cos x u2 = ln | sec x + tan x| 124 3.5 and y = c1 cos x + c2 sin x − cos x sec x + sin x ln | sec x + tan x| = c1 cos x + c2 sin x − 1 + sin x ln | sec x + tan x|. 7. The auxiliary equation is m2 − 1 = 0, so yc = c1 ex + c2 e−x and % x % %e e−x % % = −2. W = %% x e −e−x % Identifying f (x) = cosh x = 21 (e−x + ex ) we obtain u′1 = 1 −2x 1 e + 4 4 1 1 u′2 = − − e2x . 4 4 Then 1 1 u1 = − e−2x + x 8 4 1 1 u2 = − e2x − x 8 4 and 1 1 1 1 y = c1 ex + c2 e−x − e−x + xex − ex − xe−x 8 4 8 4 1 = c3 ex + c4 e−x + x(ex − e−x ) 4 1 = c3 ex + c4 e−x + x sinh x. 2 8. The auxiliary equation is m2 − 1 = 0, so yc = c1 ex + c2 e−x and % % x %e e−x %% = −2. W = %% x e −e−x % Identifying f (x) = sinh 2x we obtain 1 1 u′1 = − e−3x + ex 4 4 u′2 = Then u1 = 1 −x 1 3x e − e . 4 4 1 −3x 1 x e + e 12 4 1 1 u2 = − e−x − e3x . 4 12 and 1 1 −2x 1 2x 1 −2x e + e − e − e2x 12 4 4 12 ! " 1 = c1 ex + c2 e−x + e2x − e−2x 6 y = c1 ex + c2 e−x + = c1 ex + c2 e−x + 1 sinh 2x. 3 9. The auxiliary equation is m2 − 4 = 0, so yc = c1 e2x + c2 e−2x and % 2x % % e e−2x % % = −4. W = %% 2x 2e −2e−2x % 125 Variation of Parameters 3.5 Variation of Parameters Identifying f (x) = e2x /x we obtain u′1 = 1/4x and u′2 = −e4x /4x. Then 1 ln |x|, 4 ' 1 x e4t u2 = − dt 4 x0 t u1 = and 2x y = c1 e −2x + c2 e 1 + 4 ( 2x e −2x ln |x| − e ' x x0 ) e4t dt , t x0 > 0. 10. The auxiliary equation is m2 − 9 = 0, so yc = c1 e3x + c2 e−3x and % 3x % % e e−3x % % = −6. W = %% 3x 3e −3e−3x % Identifying f (x) = 9x/e3x we obtain u′1 = 23 xe−6x and u′2 = − 32 x. Then 1 −6x 1 −6x e − xe , 24 4 3 u2 = − x2 4 u1 = − and y = c1 e3x + c2 e−3x − 1 −3x 1 −3x 3 2 −3x e − xe − x e 24 4 4 1 = c1 e3x + c3 e−3x − xe−3x (1 − 3x). 4 11. The auxiliary equation is m2 + 3m + 2 = (m + 1)(m + 2) = 0, so yc = c1 e−x + c2 e−2x and % −x % % e e−2x % % % = −e−3x . W = % −x −e −2e−2x % Identifying f (x) = 1/(1 + ex ) we obtain u′1 = ex 1 + ex u′2 = − ex e2x = − ex . x 1+e 1 + ex Then u1 = ln(1 + ex ), u2 = ln(1 + ex ) − ex , and y = c1 e−x + c2 e−2x + e−x ln(1 + ex ) + e−2x ln(1 + ex ) − e−x = c3 e−x + c2 e−2x + (1 + e−x )e−x ln(1 + ex ). 12. The auxiliary equation is m2 − 2m + 1 = (m − 1)2 = 0, so yc = c1 ex + c2 xex and % % x %e xex % % % = e2x . W =% x e xex + ex % " ! Identifying f (x) = ex / 1 + x2 we obtain u′1 = − u′2 = xex ex x =− 2x 2 e (1 + x ) 1 + x2 e2x ex ex 1 = . (1 + x2 ) 1 + x2 126 3.5 Variation of Parameters " ! Then u1 = − 21 ln 1 + x2 , u2 = tan−1 x, and " ! 1 y = c1 ex + c2 xex − ex ln 1 + x2 + xex tan−1 x. 2 13. The auxiliary equation is m2 + 3m + 2 = (m + 1)(m + 2) = 0, so yc = c1 e−x + c2 e−2x and % −x % % e e−2x % % = −e−3x . W = %% −x −e −2e−2x % Identifying f (x) = sin ex we obtain u′1 = e−2x sin ex = ex sin ex e−3x e−x sin ex = −e2x sin ex . −e−3x Then u1 = − cos ex , u2 = ex cos x − sin ex , and u′2 = y = c1 e−x + c2 e−2x − e−x cos ex + e−x cos ex − e−2x sin ex = c1 e−x + c2 e−2x − e−2x sin ex . 14. The auxiliary equation is m2 − 2m + 1 = (m − 1)2 = 0, so yc = c1 et + c2 tet and % t % %e tet % % % = e2t . W =% t e tet + et % Identifying f (t) = et tan−1 t we obtain u′1 = − u′2 = Then tet et tan−1 t = −t tan−1 t e2t et et tan−1 t = tan−1 t. e2t 1 + t2 t tan−1 t + 2 2 " 1 ! u2 = t tan−1 t − ln 1 + t2 2 u1 = − and ( ) ( ) " 1 + t2 t 1 ! y = c1 et + c2 tet + − tan−1 t + et + t tan−1 t − ln 1 + t2 tet 2 2 2 " ! ". 1 -! = c1 et + c3 tet + et t2 − 1 tan−1 t − ln 1 + t2 . 2 15. The auxiliary equation is m2 + 2m + 1 = (m + 1)2 = 0, so yc = c1 e−t + c2 te−t and % % −t % % e te−t % = e−2t . % W = % −t −t −t % −e −te + e Identifying f (t) = e−t ln t we obtain u′1 = − u′2 = te−t e−t ln t = −t ln t e−2t e−t e−t ln t = ln t. e−2t 127 3.5 Variation of Parameters Then 1 1 u1 = − t2 ln t + t2 2 4 u2 = t ln t − t and 1 y = c1 e−t + c2 te−t − t2 e−t ln t + 2 1 = c1 e−t + c2 te−t + t2 e−t ln t − 2 1 2 −t t e + t2 e−t ln t − t2 e−t 4 3 2 −t t e . 4 16. The auxiliary equation is 2m2 + 2m + 1 = 0, so yc = e−x/2 [c1 cos(x/2) + c2 sin(x/2)] and % x % e−x/2 cos % 2 % W =% % 1 −x/2 x 1 x %− e cos − e−x/2 sin 2 2 2 2 x 2 % % % 1 % % = e−x . x 1 x/2 x% 2 1 −x/2 e cos − e sin % 2 2 2 2 e−x/2 sin √ Identifying f (x) = 2 x we obtain √ √ e−x/2 sin(x/2)2 x x = −4ex/2 x sin −x/2 2 e √ √ x e−x/2 cos(x/2)2 x u′2 = − = 4ex/2 x cos . −x/2 2 e u′1 = − Then ' x x x$ x − 4e−x/2 cos c1 cos + c2 sin 2 2 2 ' √ t et/2 t sin dt 2 x0 ' x √ t et/2 t cos dt u2 = 4 2 x0 u1 = −4 and −x/2 y=e # x t/2 e x0 √ x t t sin dt + 4e−x/2 sin 2 2 ' x0 17. The auxiliary equation is 3m2 − 6m + 6 = 0, so yc = ex (c1 cos x + c2 sin x) and % % % % ex cos x ex sin x % % = e2x . W =% x x x x e cos x − e sin x e cos x + e sin x % Identifying f (x) = 31 ex sec x we obtain u′1 = − u′2 = Then u1 = 1 3 (ex sin x)(ex sec x)/3 1 = − tan x e2x 3 1 (ex cos x)(ex sec x)/3 = . e2x 3 ln(cos x), u2 = 13 x, and y = c1 ex cos x + c2 ex cos x + 1 1 ln(cos x)ex cos x + xex sin x. 3 3 18. The auxiliary equation is 4m2 − 4m + 1 = (2m − 1)2 = 0, so yc = c1 ex/2 + c2 xex/2 and % x/2 % % e % xex/2 % = ex . W = %% 1 x/2 1 x/2 x/2 % e xe + e 2 2 128 x √ t et/2 t cos dt. 2 3.5 Variation of Parameters √ Identifying f (x) = 41 ex/2 1 − x2 we obtain √ xex/2 ex/2 1 − x2 1 3 = − x 1 − x2 x 4e 4 √ 13 ex/2 ex/2 1 − x2 = 1 − x2 . u′2 = 4ex 4 To find u1 and u2 we use the substitution v = 1 − x2 and the trig substitution x = sin θ, respectively: "3/2 1 ! u1 = 1 − x2 12 x3 1 u2 = 1 − x2 + sin−1 x. 8 8 Thus ! "3/2 1 2 x/2 3 1 1 + x e y = c1 ex/2 + c2 xex/2 + ex/2 1 − x2 1 − x2 + xex/2 sin−1 x. 12 8 8 u′1 = − 19. The auxiliary equation is 4m2 − 1 = (2m − 1)(2m + 1) = 0, so yc = c1 ex/2 + c2 e−x/2 and % x/2 % % e e−x/2 % % = −1. W = %% 1 x/2 − 21 e−x/2 % 2e Identifying f (x) = xex/2 /4 we obtain u′1 = x/4 and u′2 = −xex /4. Then u1 = x2 /8 and u2 = −xex /4 + ex /4. Thus 1 y = c1 ex/2 + c2 e−x/2 + x2 ex/2 − 8 1 = c3 ex/2 + c2 e−x/2 + x2 ex/2 − 8 and y′ = 1 x/2 1 x/2 xe + e 4 4 1 x/2 xe 4 1 1 1 1 x/2 1 −x/2 c3 e − c2 e + x2 ex/2 + xex/2 − ex/2 . 2 2 16 8 4 The initial conditions imply c3 + c2 =1 1 1 1 c3 − c2 − = 0. 2 2 4 Thus c3 = 3/4 and c2 = 1/4, and y= 3 x/2 1 −x/2 1 2 x/2 1 x/2 e + e + x e − xe . 4 4 8 4 20. The auxiliary equation is 2m2 + m − 1 = (2m − 1)(m + 1) = 0, so yc = c1 ex/2 + c2 e−x and % x/2 % % e e−x % % % = − 3 e−x/2 . W = % 1 x/2 −x % 2 −e 2e Identifying f (x) = (x + 1)/2 we obtain 1 −x/2 (x + 1) e 3 1 u′2 = − ex (x + 1). 3 u′1 = Then u1 = −e−x/2 ( 1 u2 = − xex . 3 129 ) 2 x−2 3 3.5 Variation of Parameters Thus y = c1 ex/2 + c2 e−x − x − 2 and y′ = 1 x/2 − c2 e−x − 1. c1 e 2 The initial conditions imply c1 − c2 − 2 = 1 1 c1 − c2 − 1 = 0. 2 Thus c1 = 8/3 and c2 = 1/3, and y= 8 x/2 1 −x e + e − x − 2. 3 3 21. The auxiliary equation is m2 + 2m − 8 = (m − 2)(m + 4) = 0, so yc = c1 e2x + c2 e−4x and % 2x % % e e−4x % % % = −6e−2x . W = % 2x 2e −4e−4x % Identifying f (x) = 2e−2x − e−x we obtain 1 −4x 1 −3x e − e 3 6 1 1 u′2 = e3x − e2x . 6 3 u′1 = Then 1 −4x 1 e + e−3x 12 18 1 3x 1 2x u2 = e − e . 18 6 u1 = − Thus 1 1 1 1 −2x e + e−x + e−x − e−2x 12 18 18 6 1 1 = c1 e2x + c2 e−4x − e−2x + e−x 4 9 y = c1 e2x + c2 e−4x − and 1 1 y ′ = 2c1 e2x − 4c2 e−4x + e−2x − e−x . 2 9 The initial conditions imply 5 =1 36 7 2c1 − 4c2 + = 0. 18 c1 + c2 − Thus c1 = 25/36 and c2 = 4/9, and y= 25 2x 4 −4x 1 −2x 1 −x e + e − e + e . 36 9 4 9 22. The auxiliary equation is m2 − 4m + 4 = (m − 2)2 = 0, so yc = c1 e2x + c2 xe2x and % 2x % % e % xe2x % = e4x . W = %% 2x 2x 2x % 2e 2xe + e 130 3.5 ! " Identifying f (x) = 12x2 − 6x e2x we obtain Variation of Parameters u′1 = 6x2 − 12x3 u′2 = 12x2 − 6x. Then u1 = 2x3 − 3x4 u2 = 4x3 − 3x2 . Thus and ! " ! " y = c1 e2x + c2 xe2x + 2x3 − 3x4 e2x + 4x3 − 3x2 xe2x ! " = c1 e2x + c2 xe2x + e2x x4 − x3 " ! " ! " ! y ′ = 2c1 e2x + c2 2xe2x + e2x + e2x 4x3 − 3x2 + 2e2x x4 − x3 . The initial conditions imply c1 =1 2c1 + c2 = 0. Thus c1 = 1 and c2 = −2, and ! " ! " y = e2x − 2xe2x + e2x x4 − x3 = e2x x4 − x3 − 2x + 1 . 23. Write the equation in the form y ′′ + ( ) 1 1 ′ y + 1 − 2 y = x−1/2 x 4x and identify f (x) = x−1/2 . From y1 = x−1/2 cos x and y2 = x−1/2 sin x we compute % % x−1/2 cos x W (y1 , y2 ) = %% −x−1/2 sin x − 1 x−3/2 cos x 2 Now u′1 = − sin x so % % 1 x−1/2 sin x %= . 1 −3/2 −1/2 x cos x − 2 x sin x % x u1 = cos x, and u′2 = cos x so u2 = sin x. Thus a particular solution is yp = x−1/2 cos2 x + x−1/2 sin2 x, and the general solution is y = c1 x−1/2 cos x + c2 x−1/2 sin x + x−1/2 cos2 x + x−1/2 sin2 x = c1 x−1/2 cos x + c2 x−1/2 sin x + x−1/2 . 24. Write the equation in the form 1 ′ 1 sec(ln x) y + 2y = x x x2 2 and identify f (x) = sec(ln x)/x . From y1 = cos(ln x) and y2 = sin(ln x) we compute % % % cos(ln x) sin(ln x) % % 1 % %= . W = %% % x cos(ln x) sin(ln x) % %− x x y ′′ + 131 3.5 Variation of Parameters Now u′1 = − tan(ln x) x so u1 = ln | cos(ln x)|, 1 x so u2 = ln x. and u′2 = Thus, a particular solution is yp = cos(ln x) ln | cos(ln x)| + (ln x) sin(ln x), and the general solution is y = c1 cos(ln x) + c2 sin(ln x) + cos(ln x) ln | cos(ln x)| + (ln x) sin(ln x). 25. The auxiliary equation is m3 + m = m(m2 + 1) = 0, so yc = c1 + c2 cos x + c3 sin x and % % %1 cos x sin x %% % % % W = % 0 − sin x cos x % = 1. % % % 0 − cos x − sin x % Identifying f (x) = tan x we obtain % % 0 cos x % % ′ − sin x u1 = W 1 = % 0 % % tan x − cos x % %1 0 % % ′ u2 = W2 = % 0 0 % % 0 tan x Then % sin x %% % cos x % = tan x % − sin x % % sin x %% % cos x % = − sin x % − sin x % % % %1 cos x 0 %% % cos2 x − 1 % % = cos x − sec x. 0 % = − sin x tan x = u′3 = W3 = % 0 − sin x % % cos x % 0 − cos x tan x % u1 = − ln | cos x| u2 = cos x u3 = sin x − ln | sec x + tan x| and y = c1 + c2 cos x + c3 sin x − ln | cos x| + cos2 x + sin2 x − sin x ln | sec x + tan x| = c4 + c2 cos x + c3 sin x − ln | cos x| − sin x ln | sec x + tan x| for −π/2 < x < π/2. ! " 26. The auxiliary equation is m3 + 4m = m m2 + 4 = 0, so yc = c1 + c2 cos 2x + c3 sin 2x and % % %1 cos 2x sin 2x %% % % % 2 cos 2x % = 8. W = % 0 −2 sin 2x % % % 0 −4 cos 2x −4 sin 2x % 132 3.5 Variation of Parameters Identifying f (x) = sec 2x we obtain % % % 0 cos 2x sin 2x %% % 1 1% % 1 −2 sin 2x 2 cos 2x % = sec 2x u′1 = W1 = % 0 % 4 8 8% % sec 2x −4 cos 2x −4 sin 2x % % % %1 0 sin 2x %% % 1% 1 1 % 0 2 cos 2x % = − u′2 = W2 = % 0 % 8 8% 4 % 0 sec 2x −4 sin 2x % % % %1 cos 2x 0 %% % 1 1 1% % 0 % = − tan 2x. u′3 = W3 = % 0 −2 sin 2x % 8 8% 4 % 0 −4 cos 2x sec 2x % Then 1 ln | sec 2x + tan 2x| 8 1 u2 = − x 4 1 u3 = ln | cos 2x| 8 u1 = and y = c1 + c2 cos 2x + c3 sin 2x + 1 1 1 ln | sec 2x + tan 2x| − x cos 2x + sin 2x ln | cos 2x| 8 4 8 for −π/4 < x < π/4. 27. The auxiliary equation is 3m2 −6m+30 = 0, which has roots 1±3i, so yc = ex (c1 cos 3x+c2 sin 3x). We consider first the differential equation 3y ′′ − 6y ′ + 30y = 15 sin x, which can be solved using undetermined coefficients. Letting yp1 = A cos x + B sin x and substituting into the differential equation we get (27A − 6B) cos x + (6A + 27B) sin x = 15 sin x. Then 27A − 6B = 0 and 6A + 27B = 15, 2 17 2 9 9 and B = 17 . Thus, yp1 = 17 cos x+ 17 sin x. Next, we consider the differential equation 3y ′′ −6y ′ +30y, for which a particular solution yp2 can be found using variation of parameters. The Wronskian is % % % % ex cos 3x ex sin 3x % % = 3e2x . W =% x x x x e cos 3x − 3e sin 3x 3e cos 3x + e sin 3x % so A = Identifying f (x) = 13 ex tan x we obtain 1 1 u′1 = − sin 3x tan 3x = − 9 9 so u1 = − Next u′2 = Thus ( sin2 3x cos 3x ) =− 1 9 ( 1 − cos2 3x cos 3x ) 1 = − (sec 3x − cos 3x) 9 1 1 ln | sec 3x + tan 3x| + sin 3x. 27 27 1 sin 3x so 9 u2 = − 1 cos 3x. 27 1 x 1 x e cos 3x(ln | sec 3x + tan 3x| − sin 3x) − e sin 3x cos 3x 27 27 1 = − ex (cos 3x) ln | sec 3x + tan 3x| 27 yp2 = − 133 3.5 Variation of Parameters and the general solution of the original differential equation is y = ex (c1 cos 3x + c2 sin 3x) + yp1 (x) + yp2 (x). 28. The auxiliary equation is m2 − 2m + 1 = (m − 1)2 = 0, which has repeated root 1, so yc = c1 ex + c2 xex . We consider first the differential equation y ′′ − 2y ′ + y = 4x2 − 3, which can be solved using undetermined coefficients. Letting yp1 = Ax2 + Bx + C and substituting into the differential equation we get Ax2 + (−4A + B)x + (2A − 2B + C) = 4x2 − 3. Then A = 4, −4A + B = 0, and 2A − 2B + C = −3, so A = 4, B = 16, and C = 21. Thus, yp1 = 4x2 + 16x + 21. Next we consider the differential equation y ′′ − 2y ′ + y = x−1 ex , for which a particular solution yp2 can be found using variation of parameters. The Wronskian is % x %e W = %% x e % % % = e2x . x x% xe + e xex Identifying f (x) = ex /x we obtain u′1 = −1 and u′2 = 1/x. Then u1 = −x and u2 = ln x, so that yp2 = −xex + xex ln x, and the general solution of the original differential equation is y = yc + yp1 + yp2 = c1 ex + c2 xex + 4x2 + 16x + 21 − xex + xex ln x = c1 ex + c3 xex + 4x2 + 16x + 21 + xex ln x . 29. The interval of definition for Problem 1 is (−π/2, π/2), for Problem 7 is (−∞, ∞), for Problem 9 is (0, ∞), and for Problem 18 is (−1, 1). In Problem 24 the general solution is y = c1 cos(ln x) + c2 sin(ln x) + cos(ln x) ln | cos(ln x)| + (ln x) sin(ln x) for −π/2 < ln x < π/2 or e−π/2 < x < eπ/2 . The bounds on ln x are due to the presence of sec(ln x) in the differential equation. 30. We are given that y1 = x2 is a solution of x4 y ′′ + x3 y ′ − 4x2 y = 0. To find a second solution we use reduction of order. Let y = x2 u(x). Then the product rule gives y ′ = x2 u′ + 2xu and y ′′ = x2 u′′ + 4xu′ + 2u, so x4 y ′′ + x3 y ′ − 4x2 y = x5 (xu′′ + 5u′ ) = 0. Letting w = u′ , this becomes xw′ + 5w = 0. Separating variables and integrating we have dw 5 = − dx w x and ln |w| = −5 ln x + c. Thus, w = x−5 and u = − 41 x−4 . A second solution is then y2 = x2 x−4 = 1/x2 , and the general solution of the homogeneous differential equation is yc = c1 x2 + c2 /x2 . To find a particular solution, yp , we use variation of parameters. The Wronskian is % 2 % %x 1/x2 % % % = −4 . W =% 3% x 2x −2/x 1 −4 Identifying f (x) = 1/x4 we obtain u′1 = 41 x−5 and u′2 = − 14 x−1 . Then u1 = − 16 x and u2 = − 14 ln x, so yp = − 1 1 1 −4 2 1 x x − (ln x)x−2 = − x−2 − x−2 ln x. 16 4 16 4 134 3.6 Cauchy-Euler Equation The general solution is y = c1 x2 + c2 1 1 − − 2 ln x. x2 16x2 4x EXERCISES 3.6 Cauchy-Euler Equation 1. The auxiliary equation is m2 − m − 2 = (m + 1)(m − 2) = 0 so that y = c1 x−1 + c2 x2 . 2. The auxiliary equation is 4m2 − 4m + 1 = (2m − 1)2 = 0 so that y = c1 x1/2 + c2 x1/2 ln x. 3. The auxiliary equation is m2 = 0 so that y = c1 + c2 ln x. 4. The auxiliary equation is m2 − 4m = m(m − 4) = 0 so that y = c1 + c2 x4 . 5. The auxiliary equation is m2 + 4 = 0 so that y = c1 cos(2 ln x) + c2 sin(2 ln x). 6. The auxiliary equation is m2 + 4m + 3 = (m + 1)(m + 3) = 0 so that y = c1 x−1 + c2 x−3 . √ 7. The auxiliary equation is m2 − 4m − 2 = 0 so that y = c1 x2− 6 + c2 x2+ √ √ 6 . √ 8. The auxiliary equation is m2 + 2m − 4 = 0 so that y = c1 x−1+ 5 + c2 x−1− 5 . " ! " ! 9. The auxiliary equation is 25m2 + 1 = 0 so that y = c1 cos 51 ln x + c2 sin 51 ln x . 10. The auxiliary equation is 4m2 − 1 = (2m − 1)(2m + 1) = 0 so that y = c1 x1/2 + c2 x−1/2 . 11. The auxiliary equation is m2 + 4m + 4 = (m + 2)2 = 0 so that y = c1 x−2 + c2 x−2 ln x. 12. The auxiliary equation is m2 + 7m + 6 = (m + 1)(m + 6) = 0 so that y = c1 x−1 + c2 x−6 . 13. The auxiliary equation is 3m2 + 3m + 1 = 0 so that 5 /√ 0 06 /√ 3 3 −1/2 c1 cos y=x ln x + c2 sin ln x . 6 6 14. The auxiliary equation is m2 − 8m + 41 = 0 so that y = x4 [c1 cos(5 ln x) + c2 sin(5 ln x)]. 15. Assuming that y = xm and substituting into the differential equation we obtain m(m − 1)(m − 2) − 6 = m3 − 3m2 + 2m − 6 = (m − 3)(m2 + 2) = 0. Thus y = c1 x3 + c2 cos #√ $ #√ $ 2 ln x + c3 sin 2 ln x . 16. Assuming that y = xm and substituting into the differential equation we obtain m(m − 1)(m − 2) + m − 1 = m3 − 3m2 + 3m − 1 = (m − 1)3 = 0. Thus y = c1 x + c2 x ln x + c3 x(ln x)2 . 135 3.6 Cauchy-Euler Equation 17. Assuming that y = xm and substituting into the differential equation we obtain m(m − 1)(m − 2)(m − 3) + 6m(m − 1)(m − 2) = m4 − 7m2 + 6m = m(m − 1)(m − 2)(m + 3) = 0. Thus y = c1 + c2 x + c3 x2 + c4 x−3 . 18. Assuming that y = xm and substituting into the differential equation we obtain m(m − 1)(m − 2)(m − 3) + 6m(m − 1)(m − 2) + 9m(m − 1) + 3m + 1 = m4 + 2m2 + 1 = (m2 + 1)2 = 0. Thus y = c1 cos(ln x) + c2 sin(ln x) + c3 (ln x) cos(ln x) + c4 (ln x) sin(ln x). 19. The auxiliary equation is m2 − 5m = m(m − 5) = 0 so that yc = c1 + c2 x5 and % % % 1 x5 % 5 % = 5x4 . % W (1, x ) = % 0 5x4 % 1 5 Identifying f (x) = x3 we obtain u′1 = − 15 x4 and u′2 = 1/5x. Then u1 = − 25 x , u2 = y = c1 + c2 x5 − 1 5 ln x, and 1 5 1 5 1 x + x ln x = c1 + c3 x5 + x5 ln x. 25 5 5 20. The auxiliary equation is 2m2 + 3m + 1 = (2m + 1)(m + 1) = 0 so that yc = c1 x−1 + c2 x−1/2 and % −1 % % x x−1/2 % 1 −5/2 −1 −1/2 % %= x W (x , x )=% . −x−2 − 12 x−3/2 % 2 Identifying f (x) = 1 2 − 1 2x u2 = 25 x5/2 − 32 x3/2 , and we obtain u′1 = x − x2 and u′2 = x3/2 − x1/2 . Then u1 = 12 x2 − 31 x3 , 1 1 2 2 1 1 y = c1 x−1 + c2 x−1/2 + x − x2 + x2 − x = c1 x−1 + c2 x−1/2 − x + x2 . 2 3 5 3 6 15 21. The auxiliary equation is m2 − 2m + 1 = (m − 1)2 = 0 so that yc = c1 x + c2 x ln x and % % % x x ln x % % % = x. W (x, x ln x) = % 1 1 + ln x % Identifying f (x) = 2/x we obtain u′1 = −2 ln x/x and u′2 = 2/x. Then u1 = −(ln x)2 , u2 = 2 ln x, and y = c1 x + c2 x ln x − x(ln x)2 + 2x(ln x)2 = c1 x + c2 x ln x + x(ln x)2 , x > 0. 22. The auxiliary equation is m2 − 3m + 2 = (m − 1)(m − 2) = 0 so that yc = c1 x + c2 x2 and % % % x x2 % 2 % % = x2 . W (x, x ) = % 1 2x % Identifying f (x) = x2 ex we obtain u′1 = −x2 ex and u′2 = xex . Then u1 = −x2 ex + 2xex − 2ex , u2 = xex − ex , and y = c1 x + c2 x2 − x3 ex + 2x2 ex − 2xex + x3 ex − x2 ex = c1 x + c2 x2 + x2 ex − 2xex . 23. The auxiliary equation m(m − 1) + m − 1 = m2 − 1 = 0 has roots m1 = −1, m2 = 1, so yc = c1 x−1 + c2 x. With y1 = x−1 , y2 = x, and the identification f (x) = ln x/x2 , we get W = 2x−1 , W1 = − ln x/x, 136 and W2 = ln x/x3 . 3.6 Cauchy-Euler Equation Then u′1 = W1 /W = −(ln x)/2, u′2 = W2 /W = (ln x)/2x2 , and integration by parts gives 1 1 x − x ln x 2 2 1 1 −1 u2 = − x ln x − x−1 , 2 2 u1 = so yp = u 1 y 1 + u 2 y 2 = ( ) ( ) 1 1 −1 1 −1 1 −1 x − x ln x x + − x ln x − x x = − ln x 2 2 2 2 and y = yc + yp = c1 x−1 + c2 x − ln x, x > 0. 24. The auxiliary equation m(m − 1) + m − 1 = m2 − 1 = 0 has roots m1 = −1, m2 = 1, so yc = c1 x−1 + c2 x. With y1 = x−1 , y2 = x, and the identification f (x) = 1/x2 (x + 1), we get W = 2x−1 , Then u′1 = W1 /W = −1/2(x + 1), gives W1 = −1/x(x + 1), and W2 = 1/x3 (x + 1). u′2 = W2 /W = 1/2x2 (x + 1), and integration (by partial fractions for u′2 ) 1 u1 = − ln(x + 1) 2 1 1 −1 1 u2 = − x − ln x + ln(x + 1), 2 2 2 so + * + * 1 −1 1 1 1 −1 yp = u1 y1 + u2 y2 = − ln(x + 1) x + − x − ln x + ln(x + 1) x 2 2 2 2 ( ) 1 1 1 ln(x + 1) 1 1 1 ln(x + 1) = − − x ln x + x ln(x + 1) − = − + x ln 1 + − 2 2 2 2x 2 2 x 2x and y = yc + yp = c1 x−1 + c2 x − ( ) 1 1 1 ln(x + 1) + x ln 1 + , − 2 2 x 2x x > 0. y 25. The auxiliary equation is m2 + 2m = m(m + 2) = 0, so that y = c1 + c2 x−2 and y ′ = −2c2 x−3 . The initial conditions imply 5x c1 + c2 = 0 −2c2 = 4. Thus, c1 = 2, c2 = −2, and y = 2 − 2x−2 . The graph is given to the right. -10 -20 137 3.6 Cauchy-Euler Equation 26. The auxiliary equation is m2 − 6m + 8 = (m − 2)(m − 4) = 0, so that 2 4 y = c1 x + c2 x y 3 ′ and y = 2c1 x + 4c2 x . 30 The initial conditions imply 20 4c1 + 16c2 = 32 10 4c1 + 32c2 = 0. -1 Thus, c1 = 16, c2 = −2, and y = 16x2 − 2x4 . The graph is given to the right. 4 x -4 -20 -30 27. The auxiliary equation is m2 + 1 = 0, so that y 3 y = c1 cos(ln x) + c2 sin(ln x) and 1 1 sin(ln x) + c2 cos(ln x). x x The initial conditions imply c1 = 1 and c2 = 2. Thus y ′ = −c1 100 x 50 -3 y = cos(ln x) + 2 sin(ln x). The graph is given to the right. 28. The auxiliary equation is m2 − 4m + 4 = (m − 2)2 = 0, so that y y = c1 x2 + c2 x2 ln x and y ′ = 2c1 x + c2 (x + 2x ln x). 5 The initial conditions imply c1 = 5 and c2 + 10 = 3. Thus y = 5x2 − 7x2 ln x. The graph is given to the right. x -10 -20 -30 29. The auxiliary equation is m2 = 0 so that yc = c1 + c2 ln x and % % % 1 ln x % 1 % %= . W (1, ln x) = % 0 1/x % x y 15 Identifying f (x) = 1 we obtain u′1 = −x ln x and u′2 = x. Then u1 = 41 x2 − 12 x2 ln x, u2 = 21 x2 , and 10 1 1 1 1 y = c1 + c2 ln x + x2 − x2 ln x + x2 ln x = c1 + c2 ln x + x2 . 4 2 2 4 The initial conditions imply c1 + y= 3 4 − ln x + 1 2 4x . 1 4 = 1 and c2 + 1 2 The graph is given to the right. = − 21 . Thus, c1 = 3 4 5 , c2 = −1, and 5 138 3.6 30. The auxiliary equation is m2 − 6m + 8 = (m − 2)(m − 4) = 0, so that yc = c1 x2 + c2 x4 and % 2 % %x x4 % % = 2x5 . W = %% 2x 4x3 % Identifying f (x) = 8x4 we obtain u′1 = −4x3 and u′2 = 4x. Then u1 = −x4 , u2 = 2x2 , and y = c1 x2 + c2 x4 + x6 . The initial conditions Cauchy-Euler Equation y 0.05 -1 1 x imply 1 1 1 c1 + c2 = − 4 16 64 3 1 c1 + c2 = − . 2 16 1 2 1 , c2 = − 21 , and y = 16 x − 12 x4 + x6 . The graph is given above. Thus c1 = 16 31. Substituting x = et into the differential equation we obtain d2 y dy +8 − 20y = 0. 2 dt dt The auxiliary equation is m2 + 8m − 20 = (m + 10)(m − 2) = 0 so that y = c1 e−10t + c2 e2t = c1 x−10 + c2 x2 . 32. Substituting x = et into the differential equation we obtain dy d2 y − 10 + 25y = 0. 2 dt dt The auxiliary equation is m2 − 10m + 25 = (m − 5)2 = 0 so that y = c1 e5t + c2 te5t = c1 x5 + c2 x5 ln x. 33. Substituting x = et into the differential equation we obtain d2 y dy +9 + 8y = e2t . 2 dt dt The auxiliary equation is m2 + 9m + 8 = (m + 1)(m + 8) = 0 so that yc = c1 e−t + c2 e−8t . Using undetermined coefficients we try yp = Ae2t . This leads to 30Ae2t = e2t , so that A = 1/30 and y = c1 e−t + c2 e−8t + 1 1 2t e = c1 x−1 + c2 x−8 + x2 . 30 30 34. Substituting x = et into the differential equation we obtain dy d2 y −5 + 6y = 2t. 2 dt dt The auxiliary equation is m2 − 5m + 6 = (m − 2)(m − 3) = 0 so that yc = c1 e2t + c2 e3t . Using undetermined coefficients we try yp = At + B. This leads to (−5A + 6B) + 6At = 2t, so that A = 1/3, B = 5/18, and 1 1 5 5 y = c1 e2t + c2 e3t + t + = c1 x2 + c2 x3 + ln x + . 3 18 3 18 35. Substituting x = et into the differential equation we obtain dy d2 y −4 + 13y = 4 + 3et . 2 dt dt 139 3.6 Cauchy-Euler Equation The auxiliary equation is m2 −4m+13 = 0 so that yc = e2t (c1 cos 3t+c2 sin 3t). Using undetermined coefficients we try yp = A + Bet . This leads to 13A + 10Bet = 4 + 3et , so that A = 4/13, B = 3/10, and 3 4 + et 13 10 4 3 = x2 [c1 cos(3 ln x) + c2 sin(3 ln x)] + + x. 13 10 y = e2t (c1 cos 3t + c2 sin 3t) + 36. From it follows that d2 y 1 = 2 dx2 x 1 x2 = 1 x2 = 1 x3 d2 y dy − dt2 dt ) ( ) ) d2 y dy 2 d2 y dy − − − dt2 dt x3 dt2 dt ) ( ( ) 1 d dy d d2 y 2 d2 y 2 dy − − 3 2 + 3 2 2 dx dt x dx dt x dt x dt ) ) ( ( 1 d2 y 1 2 d2 y 2 dy d3 y 1 − − + 3 3 2 2 3 2 dt x x dt x x dt x dt ( 3 ) d2 y dy d y − 3 +2 . dt3 dt2 dt 1 d d3 y = 2 dx3 x dx = ( ( Substituting into the differential equation we obtain ( 2 ) d3 y d y dy d2 y dy dy − 3 + 2 − 3 − − 6y = 3 + 3t +6 dt3 dt2 dt dt2 dt dt or d2 y dy d3 y − 6 + 11 − 6y = 3 + 3t. 3 2 dt dt dt The auxiliary equation is m3 −6m2 +11m−6 = (m−1)(m−2)(m−3) = 0 so that yc = c1 et +c2 e2t +c3 e3t . Using undetermined coefficients we try yp = A + Bt. This leads to (11B − 6A) − 6Bt = 3 + 3t, so that A = −17/12, B = −1/2, and 17 1 17 1 y = c1 et + c2 e2t + c3 e3t − − t = c1 x + c2 x2 + c3 x3 − − ln x. 12 2 12 2 In the next two problems we use the substitution t = −x since the initial conditions are on the interval (−∞, 0). In this case dy dx dy dy = =− dt dx dt dx and ( ) ( ) d2 y d dy d dy d dy ′ dx d2 y dx d2 y = = − = − (y ′ ) = − =− 2 = 2. 2 dt dt dt dt dx dt dx dt dx dt dx 37. The differential equation and initial conditions become % 2 % 2 d y 4t + y = 0; y(t) %% = 2, 2 dt t=1 % % y (t) %% The auxiliary equation is 4m2 − 4m + 1 = (2m − 1)2 = 0, so that y = c1 t1/2 + c2 t1/2 ln t and y ′ = ′ t=1 = −4. ) ( 1 1 −1/2 c1 t + c2 t−1/2 + t−1/2 ln t . 2 2 140 3.6 Cauchy-Euler Equation The initial conditions imply c1 = 2 and 1 + c2 = −4. Thus y = 2t1/2 − 5t1/2 ln t = 2(−x)1/2 − 5(−x)1/2 ln(−x), x < 0. 38. The differential equation and initial conditions become d2 y dy t − 4t + 6y = 0; dt2 dt 2 % % y(t) %% = 8, t=2 % % y (t) %% ′ = 0. t=2 The auxiliary equation is m2 − 5m + 6 = (m − 2)(m − 3) = 0, so that y = c1 t2 + c2 t3 and y ′ = 2c1 t + 3c2 t2 . The initial conditions imply 4c1 + 8c2 = 8 4c1 + 12c2 = 0 from which we find c1 = 6 and c2 = −2. Thus y = 6t2 − 2t3 = 6x2 + 2x3 , x < 0. 39. Letting u = x + 2 we obtain dy/dx = dy/du and, using the Chain Rule, ( ) d2 y d dy d2 y d2 y du d2 y = = = (1) = . dx2 dx du du2 dx du2 du2 Substituting into the differential equation we obtain u2 dy d2 y +u + y = 0. du2 du The auxiliary equation is m2 + 1 = 0 so that y = c1 cos(ln u) + c2 sin(ln u) = c1 cos[ ln(x + 2)] + c2 sin[ ln(x + 2)]. 40. If 1 − i is a root of the auxiliary equation then so is 1 + i, and the auxiliary equation is (m − 2)[m − (1 + i)][m − (1 − i)] = m3 − 4m2 + 6m − 4 = 0. We need m3 − 4m2 + 6m − 4 to have the form m(m − 1)(m − 2) + bm(m − 1) + cm + d. Expanding this last expression and equating coefficients we get b = −1, c = 3, and d = −4. Thus, the differential equation is x3 y ′′′ − x2 y ′′ + 3xy ′ − 4y = 0. 41. For x2 y ′′ = 0 the auxiliary equation is m(m − 1) = 0 and the general solution is y = c1 + c2 x. The initial conditions imply c1 = y0 and c2 = y1 , so y = y0 + y1 x. The initial conditions are satisfied for all real values of y0 and y1 . For x2 y ′′ − 2xy ′ + 2y = 0 the auxiliary equation is m2 − 3m + 2 = (m − 1)(m − 2) = 0 and the general solution is y = c1 x + c2 x2 . The initial condition y(0) = y0 implies 0 = y0 and the condition y ′ (0) = y1 implies c1 = y1 . Thus, the initial conditions are satisfied for y0 = 0 and for all real values of y1 . For x2 y ′′ − 4xy ′ + 6y = 0 the auxiliary equation is m2 − 5m + 6 = (m − 2)(m − 3) = 0 and the general solution is y = c1 x2 + c2 x3 . The initial conditions imply y(0) = 0 = y0 and y ′ (0) = 0. Thus, the initial conditions are satisfied only for y0 = y1 = 0. √ 42. The function y(x) = − x cos(ln x) is defined for x > 0 and has x-intercepts where ln x = π/2 + kπ for k an integer or where x = eπ/2+kπ . Solving π/2 + kπ = 0.5 we get k ≈ −0.34, so eπ/2+kπ < 0.5 for all negative integers and the graph has infinitely many x-intercepts in the interval (0, 0.5). 141 3.6 Cauchy-Euler Equation 43. The auxiliary equation is 2m(m − 1)(m − 2) − 10.98m(m − 1) + 8.5m + 1.3 = 0, so that m1 = −0.053299, m2 = 1.81164, m3 = 6.73166, and y = c1 x−0.053299 + c2 x1.81164 + c3 x6.73166 . 44. The auxiliary equation is m(m − 1)(m − 2) + 4m(m − 1) + 5m − 9 = 0, so that m1 = 1.40819 and the two complex roots are −1.20409 ± 2.22291i. The general solution of the differential equation is y = c1 x1.40819 + x−1.20409 [c2 cos(2.22291 ln x) + c3 sin(2.22291 ln x)]. 45. The auxiliary equation is m(m − 1)(m − 2)(m − 3) + 6m(m − 1)(m − 2) + 3m(m − 1) − 3m + 4 = 0, so that √ √ m1 = m2 = 2 and m3 = m4 = − 2 . The general solution of the differential equation is √ y = c1 x 2 √ + c2 x 2 √ ln x + c3 x− 2 √ + c4 x− 2 ln x. 46. The auxiliary equation is m(m − 1)(m − 2)(m − 3) − 6m(m − 1)(m − 2) + 33m(m − 1) − 105m + 169 = 0, so that m1 = m2 = 3 + 2i and m3 = m4 = 3 − 2i. The general solution of the differential equation is y = x3 [c1 cos(2 ln x) + c2 sin(2 ln x)] + x3 ln x[c3 cos(2 ln x) + c4 sin(2 ln x)]. 47. The auxiliary equation m(m − 1)(m − 2) − m(m − 1) − 2m + 6 = m3 − 4m2 + m + 6 = 0 has roots m1 = −1, m2 = 2, and m3 = 3, so yc = c1 x−1 + c2 x2 + c3 x3 . With y1 = x−1 , y2 = x2 , y3 = x3 , and the identification f (x) = 1/x, we get from (10) of Section 4.6 in the text W1 = x3 , W2 = −4, W3 = 3/x, and W = 12x. Then u′1 = W1 /W = x2 /12, u′2 = W2 /W = −1/3x, u′3 = 1/4x2 , and integration gives u1 = x3 , 36 so yp = u 1 y 1 + u 2 y 2 + u 3 y3 = and 1 u2 = − ln x, 3 and u3 = − 1 , 4x ) ) ( ( 2 1 1 1 x3 −1 x + x2 − ln x + x3 − = − x2 − x2 ln x, 36 3 4x 9 3 1 2 y = yc + yp = c1 x−1 + c2 x2 + c3 x3 − x2 − x2 ln x, 9 3 142 x > 0. 3.7 Nonlinear Equations EXERCISES 3.7 Nonlinear Equations 1. We have y1′ = y1′′ = ex , so (y1′′ )2 = (ex )2 = e2x = y12 . Also, y2′ = − sin x and y2′′ = − cos x, so (y2′′ )2 = (− cos x)2 = cos2 x = y22 . However, if y = c1 y1 + c2 y2 , we have (y ′′ )2 = (c1 ex − c2 cos x)2 and y 2 = (c1 ex + c2 cos x)2 . Thus (y ′′ )2 ̸= y 2 . 2. We have y1′ = y1′′ = 0, so y1 y1′′ = 1 · 0 = 0 = Also, y2′ = 2x and y2′′ = 2, so 1 2 1 (0) = (y1′ )2 . 2 2 1 1 (2x)2 = (y2′ )2 . 2 2 2 = (c1 · 1 + c2 x )(c1 · 0 + 2c2 ) = 2c2 (c1 + c2 x2 ) and y2 y2′′ = x2 (2) = 2x2 = However, if y = c1 y1 + c2 y2 , we have yy ′′ 1 2 [c1 2 · 0 + c2 (2x)] = 2c22 x2 . ′′ Thus yy ̸= 1 ′ 2 2 (y ) = 1 ′ 2 2 (y ) . 3. Let u = y ′ so that u′ = y ′′ . The equation becomes u′ = −u − 1 which is separable. Thus du = −dx =⇒ tan−1 u = −x + c1 =⇒ y ′ = tan(c1 − x) =⇒ y = ln | cos(c1 − x)| + c2 . u2 + 1 4. Let u = y ′ so that u′ = y ′′ . The equation becomes u′ = 1 + u2 . Separating variables we obtain du = dx =⇒ tan−1 u = x + c1 =⇒ u = tan(x + c1 ) =⇒ y = − ln | cos(x + c1 )| + c2 . 1 + u2 5. Let u = y ′ so that u′ = y ′′ . The equation becomes x2 u′ + u2 = 0. Separating variables we obtain ) ( ) ( x 1 dx 1 1 c1 x + 1 1 1 du = = − =⇒ − = + c = =⇒ u = − − 1 1 u2 x2 u x x c1 x + 1/c1 c1 c1 x + 1 1 1 =⇒ y = 2 ln |c1 x + 1| − x + c2 . c1 c1 6. Let u = y ′ so that y ′′ = u du/dy. The equation becomes (y + 1)u du/dy = u2 . Separating variables we obtain dy du = =⇒ ln |u| = ln |y + 1| + ln c1 =⇒ u = c1 (y + 1) u y+1 dy dy =⇒ = c1 (y + 1) =⇒ = c1 dx dx y+1 =⇒ ln |y + 1| = c1 x + c2 =⇒ y + 1 = c3 ec1 x . 7. Let u = y ′ so that y ′′ = u du/dy. The equation becomes u du/dy + 2yu3 = 0. Separating variables we obtain du 1 1 1 =⇒ y ′ = 2 + 2y dy = 0 =⇒ − + y 2 = c =⇒ u = 2 u2 u y + c1 y + c1 ! " 1 =⇒ y 2 + c1 dy = dx =⇒ y 3 + c1 y = x + c2 . 3 143 3.7 Nonlinear Equations 8. Let u = y ′ so that y ′′ = u du/dy. The equation becomes y 2 u du/dy = u. Separating variables we obtain du = 1 c1 y − 1 y dy =⇒ u = − + c1 =⇒ y ′ = =⇒ dy = dx y2 y y c1 y − 1 ( ) 1 1 1 1 y + 2 ln |y − 1| = x + c2 . 1+ dy = dx (for c1 ̸= 0) =⇒ =⇒ c1 c1 y − 1 c1 c1 If c1 = 0, then y dy = −dx and another solution is 12 y 2 = −x + c2 . y 9. (a) 10 −π/2 x 3π/2 −10 (b) Let u = y ′ so that y ′′ = u du/dy. The equation becomes u du/dy + yu = 0. Separating variables we obtain 1 1 du = −y dy =⇒ u = − y 2 + c1 =⇒ y ′ = − y 2 + c1 . 2 2 When x = 0, y = 1 and y ′ = −1 so −1 = −1/2 + c1 and c1 = −1/2. Then dy 1 1 dy 1 1 = − y2 − =⇒ 2 = − dx =⇒ tan−1 y = − x + c2 dx 2 2 y +1 2 2 ( ) 1 =⇒ y = tan − x + c2 . 2 When x = 0, y = 1 so 1 = tan c2 and c2 = π/4. The solution of the initial-value problem is ) ( π 1 − x . y = tan 4 2 The graph is shown in part (a). (c) The interval of definition is −π/2 < π/4 − x/2 < π/2 or −π/2 < x < 3π/2. 10. Let u = y ′ so that u′ = y ′′ . The equation becomes (u′ )2 + u2 = 1 √ √ which results in u′ = ± 1 − u2 . To solve u′ = 1 − u2 we separate variables: du √ = dx =⇒ sin−1 u = x + c1 =⇒ u = sin(x + c1 ) 1 − u2 −2π =⇒ y ′ = sin(x + c1 ). √ √ When x = π/2, y ′ = 3/2, so 3/2 = sin(π/2 + c1 ) and c1 = −π/6. Thus # # π$ π$ y ′ = sin x − =⇒ y = − cos x − + c2 . 6 6 144 y 2 x 2π 3.7 Nonlinear Equations When x = π/2, y = 1/2, so 1/2 = − cos(π/2−π/6)+c2 = −1/2+c2 and c2 = 1. The solution of the initial-value problem is y = 1 − cos(x − π/6). √ To solve u′ = − 1 − u2 we separate variables: du √ = −dx =⇒ cos−1 u = x + c1 1 − u2 =⇒ u = cos(x + c1 ) =⇒ y ′ = cos(x + c1 ). −2π √ √ When x = π/2, y ′ = 3/2, so 3/2 = cos(π/2 + c1 ) and c1 = −π/3. Thus # # π$ π$ =⇒ y = sin x − + c2 . y ′ = cos x − 3 3 y 1 x 2π −1 When x = π/2, y = 1/2, so 1/2 = sin(π/2 − π/3) + c2 = 1/2 + c2 and c2 = 0. The solution of the initial-value problem is y = sin(x − π/3). 11. Let u = y ′ so that u′ = y ′′ . The equation becomes u′ − (1/x)u = (1/x)u3 , which is Bernoulli. Using w = u−2 we obtain dw/dx + (2/x)w = −2/x. An integrating factor is x2 , so d 2 c1 [x w] = −2x =⇒ x2 w = −x2 + c1 =⇒ w = −1 + 2 dx x x c1 =⇒ u−2 = −1 + 2 =⇒ u = √ x c1 − x2 3 x dy =⇒ y = − =⇒ c1 − x2 + c2 =√ dx c1 − x2 =⇒ c1 − x2 = (c2 − y)2 =⇒ x2 + (c2 − y)2 = c1 . 12. Let u = y ′ so that u′ = y ′′ . The equation becomes u′ − (1/x)u = u2 , which is a Bernoulli differential equation. Using the substitution w = u−1 we obtain dw/dx + (1/x)w = −1. An integrating factor is x, so % % d 1 1 1 c1 − x2 2x [xw] = −x =⇒ w = − x + c =⇒ = =⇒ u = =⇒ y = − ln %c1 − x2 % + c2 . 2 dx 2 x u 2x c1 − x In Problems 13-16 the thinner curve is obtained using a numerical solver, while the thicker curve is the graph of the Taylor polynomial. 13. We look for a solution of the form 1 1 1 1 y(x) = y(0) + y ′ (0)x + y ′′ (0)x2 + y ′′′ (0)x3 + y (4) (0)x4 + y (5) (0)x5 . 2! 3! 4! 5! y 40 From y ′′ (x) = x + y 2 we compute y ′′′ (x) = 1 + 2yy ′ 30 y (4) (x) = 2yy ′′ + 2(y ′ )2 y (5) (x) = 2yy ′′′ + 6y ′ y ′′ . 20 Using y(0) = 1 and y ′ (0) = 1 we find y ′′ (0) = 1, y ′′′ (0) = 3, y (4) (0) = 4, y (5) (0) = 12. 10 An approximate solution is 1 1 1 1 y(x) = 1 + x + x2 + x3 + x4 + x5 . 2 2 6 10 145 0.5 1 1.5 2 2.5 3 x 3.7 Nonlinear Equations 14. We look for a solution of the form 1 1 1 1 y(x) = y(0) + y ′ (0)x + y ′′ (0)x2 + y ′′′ (0)x3 + y (4) (0)x4 + y (5) (0)x5 . 2! 3! 4! 5! y 10 From y ′′ (x) = 1 − y 2 we compute y ′′′ (x) = −2yy ′ y (4) ′′ 5 ′ 2 (x) = −2yy − 2(y ) y (5) (x) = −2yy ′′′ − 6y ′ y ′′ . 0.5 1 1.5 2 2.5 3 x ′ Using y(0) = 2 and y (0) = 3 we find y ′′ (0) = −3, y (4) (0) = −6, y ′′′ (0) = −12, y (5) (0) = 102. -5 An approximate solution is 3 1 17 y(x) = 2 + 3x − x2 − 2x3 − x4 + x5 . 2 4 20 15. We look for a solution of the form 1 1 1 1 y(x) = y(0) + y ′ (0)x + y ′′ (0)x2 + y ′′′ (0)x3 + y (4) (0)x4 + y (5) (0)x5 . 2! 3! 4! 5! From y ′′ (x) = x2 + y 2 − 2y ′ we compute -10 y 40 30 y ′′′ (x) = 2x + 2yy ′ − 2y ′′ y (4) (x) = 2 + 2(y ′ )2 + 2yy ′′ − 2y ′′′ y (5) (x) = 6y ′ y ′′ + 2yy ′′′ − 2y (4) . 20 ′ Using y(0) = 1 and y (0) = 1 we find y ′′ (0) = −1, y ′′′ (0) = 4, y (4) (0) = −6, y (5) (0) = 14. 10 An approximate solution is 1 2 1 7 y(x) = 1 + x − x2 + x3 − x4 + x5 . 2 3 4 60 0.5 1 1.5 2 2.5 3 3.5x 16. We look for a solution of the form 1 ′′ 1 1 y (0)x2 + y ′′′ (0)x3 + y (4) (0)x4 2! 3! 4! 1 (6) 1 (5) 5 6 + y (0)x + y (0)x . 5! 6! y(x) = y(0) + y ′ (0)x + From y ′′ (x) = ey we compute y 10 8 6 y ′′′ (x) = ey y ′ 4 y (4) (x) = ey (y ′ )2 + ey y ′′ 2 y (5) (x) = ey (y ′ )3 + 3ey y ′ y ′′ + ey y ′′′ y (6) (x) = ey (y ′ )4 + 6ey (y ′ )2 y ′′ + 3ey (y ′′ )2 + 4ey y ′ y ′′′ + ey y (4) . 1 -2 146 2 3 4 5x 3.7 Nonlinear Equations Using y(0) = 0 and y ′ (0) = −1 we find y ′′ (0) = 1, y ′′′ (0) = −1, y (4) (0) = 2, y (5) (0) = −5, y (6) (0) = 16. An approximate solution is 1 1 1 1 1 y(x) = −x + x2 − x3 + x4 + x5 + x6 . 2 6 12 24 45 17. We need to solve [1 + (y ′ )2 ]3/2 = y ′′ . Let u = y ′ so that u′ = y ′′ . The equation becomes (1 + u2 )3/2 = u′ or (1 + u2 )3/2 = du/dx. Separating variables and using the substitution u = tan θ we have ' ' sec2 θ du sec2 θ = dx =⇒ dθ = x dθ = x =⇒ ! " 3/2 3/2 sec3 θ (1 + u2 ) 1 + tan2 θ ' u =⇒ cos θ dθ = x =⇒ sin θ = x =⇒ √ =x 1 + u2 . y′ x2 =⇒ 3 = x =⇒ (y ′ )2 = x2 1 + (y ′ )2 = 1 − x2 1 + (y ′ )2 3 x =⇒ y ′ = √ (for x > 0) =⇒ y = − 1 − x2 . 1 − x2 18. When y = sin x, y ′ = cos x, y ′′ = − sin x, and (y ′′ )2 − y 2 = sin2 x − sin2 x = 0. When y = e−x , y ′ = −e−x , y ′′ = e−x , and (y ′′ )2 − y 2 = e−2x − e−2x = 0. From (y ′′ )2 − y 2 = 0 we have y ′′ = ±y, which can be treated as two linear equations. Since linear combinations of solutions of linear homogeneous differential equations are also solutions, we see that y = c1 ex + c2 e−x and y = c3 cos x + c4 sin x must satisfy the differential equation. However, linear combinations that involve both exponential and trigonometric functions will not be solutions since the differential equation is not linear and each type of function satisfies a different linear differential equation that is part of the original differential equation. 19. Letting u = y ′′ , separating variables, and integrating we have du 3 du = 1 + u2 , √ = dx, and dx 1 + u2 sinh−1 u = x + c1 . Then u = y ′′ = sinh(x + c1 ), y ′ = cosh(x + c1 ) + c2 , and y = sinh(x + c1 ) + c2 x + c3 . 20. If the constant −c21 is used instead of c21 , then, using partial fractions, % ) ' ( ' % x + c1 1 1 1 1 dx dx = − ln % =− y=− x2 − c21 2c1 x − c1 x + c1 2c1 % x − c1 Alternatively, the inverse hyperbolic tangent can be used. % % % +c2 . % 21. Let u = dx/dt so that d2 x/dt2 = u du/dx. The equation becomes u du/dx = −k 2 /x2 . Separating variables we obtain 1 2 k2 1 2 k2 k2 dx =⇒ u = + c =⇒ v = + c. x2 2 x 2 x When t = 0, x = x0 and v = 0 so 0 = (k 2 /x0 ) + c and c = −k 2 /x0 . Then 7 ) ( √ 1 2 1 1 dx x0 − x . v = k2 − = −k 2 and 2 x x0 dt xx0 u du = − 147 3.7 Nonlinear Equations Separating variables we have 7 ' 7 7 √ 1 x0 xx0 x − dx = k 2 dt =⇒ t = − dx. x0 − x k 2 x0 − x 22. Using Mathematica to integrate we obtain 7 * + 7 3 x0 1 x0 x −1 (x0 − 2x) t=− tan − x(x0 − x) − k 2 2 2x x0 − x 5 6 7 x0 − 2x 1 x0 3 x0 tan−1 3 . = x(x0 − x) + k 2 2 2 x(x0 − x) x x 2 x 2 10 -2 20 2 t 10 -2 x1 = 0 20 t 10 -2 x1 = 1 20 t x1 = -1.5 For d2 x/dt2 + sin x = 0 the motion appears to be periodic with amplitude 1 when x1 = 0. The amplitude and period are larger for larger magnitudes of x1 . x x 1 -1 x 1 x1 = 0 10 1 t -1 x1 = 1 10 t -1 x1 = -2.5 10 t For d2 x/dt2 + dx/dt + sin x = 0 the motion appears to be periodic with decreasing amplitude. The dx/dt term could be said to have a damping effect. EXERCISES 3.8 Linear Models: Initial-Value Problems 1. From 18 x′′ + 16x = 0 we obtain √ √ x = c1 cos 8 2 t + c2 sin 8 2 t √ √ so that the period of motion is 2π/8 2 = 2 π/8 seconds. 2. From 20x′′ + kx = 0 we obtain 1 x = c1 cos 2 so that the frequency 2/π = 1 4 7 1 k t + c2 sin 5 2 7 k t 5 3 k/5 π and k = 320 N/m. If 80x′′ + 320x = 0 then x = c1 cos 2t + c2 sin 2t so that the frequency is 2/2π = 1/π cycles/s. √ 3. From 34 x′′ + 72x = 0, x(0) = −1/4, and x′ (0) = 0 we obtain x = − 14 cos 4 6 t. 148 3.8 4. From 43 x′′ + 72x = 0, x(0) = 0, and x′ (0) = 2 we obtain x = √ 6 12 5. From 58 x′′ + 40x = 0, x(0) = 1/2, and x′ (0) = 0 we obtain x = Linear Models: Initial-Value Problems √ sin 4 6 t. 1 2 cos 8t. (a) x(π/12) = −1/4, x(π/8) = −1/2, x(π/6) = −1/4, x(π/4) = 1/2, x(9π/32) = ′ √ 2/4. ′ (b) x = −4 sin 8t so that x (3π/16) = 4 ft/s directed downward. (c) If x = 1 2 cos 8t = 0 then t = (2n + 1)π/16 for n = 0, 1, 2, . . . . 6. From 50x′′ + 200x = 0, x(0) = 0, and x′ (0) = −10 we obtain x = −5 sin 2t and x′ = −10 cos 2t. 7. From 20x′′ + 20x = 0, x(0) = 0, and x′ (0) = −10 we obtain x = −10 sin t and x′ = −10 cos t. (a) The 20 kg mass has the larger amplitude. √ (b) 20 kg: x′ (π/4) = −5 2 m/s, x′ (π/2) = 0 m/s; 50 kg: x′ (π/4) = 0 m/s, x′ (π/2) = 10 m/s (c) If −5 sin 2t = −10 sin t then 2 sin t(cos t − 1) = 0 so that t = nπ for n = 0, 1, 2, . . ., placing both masses at the equilibrium position. The 50 kg mass is moving upward; the 20 kg mass is moving upward when n is even and downward when n is odd. 8. From x′′ + 16x = 0, x(0) = −1, and x′ (0) = −2 we obtain √ 1 5 x = − cos 4t − sin 4t = cos(4t − 3.605). 2 2 √ The period is π/2 seconds and the amplitude is 5/2 feet. In 4π seconds it will make 8 complete cycles. 9. From 41 x′′ + x = 0, x(0) = 1/2, and x′ (0) = 3/2 we obtain √ 3 13 1 x = cos 2t + sin 2t = sin(2t + 0.588). 2 4 4 10. From 1.6x′′ + 40x = 0, x(0) = −1/3, and x′ (0) = 5/4 we obtain 1 1 5 x = − cos 5t + sin 5t = sin(5t − 0.927). 3 4 12 ! ! " " If x = 5/24 then t = 51 π6 + 0.927 + 2nπ and t = 15 5π 6 + 0.927 + 2nπ for n = 0, 1, 2, . . . . 11. From 2x′′ + 200x = 0, x(0) = −2/3, and x′ (0) = 5 we obtain (a) x = − 23 cos 10t + 1 2 sin 10t = 5 6 sin(10t − 0.927). (b) The amplitude is 5/6 ft and the period is 2π/10 = π/5 (c) 3π = πk/5 and k = 15 cycles. (d) If x = 0 and the weight is moving downward for the second time, then 10t − 0.927 = 2π or t = 0.721 s. (e) If x′ = 25 3 cos(10t − 0.927) = 0 then 10t − 0.927 = π/2 + nπ or t = (2n + 1)π/20 + 0.0927 for n = 0, 1, 2, . . . . (f ) x(3) = −0.597 ft (g) x′ (3) = −5.814 ft/s (h) x′′ (3) = 59.702 ft/s2 (i) If x = 0 then t = (j) If x = 5/12 then (k) If x = 5/12 and 1 ′ 10 (0.927 + nπ) for n = 0, 1, 2, . . .. The velocity at these times is x = ±8.33 ft/s. 1 1 t = 10 (π/6 + 0.927 + 2nπ) and t = 10 (5π/6 + 0.927 + 2nπ) for n = 0, 1, 2, . . . . 1 x′ < 0 then t = 10 (5π/6 + 0.927 + 2nπ) for n = 0, 1, 2, . . . . √ 12. From x′′ + 9x = 0, x(0) = −1, and x′ (0) = − 3 we obtain √ ) ( 2 4π 3 x = − cos 3t − sin 3t = √ sin 3t + 3 3 3 √ and x′ = 2 3 cos(3t + 4π/3). If x′ = 3 then t = −7π/18 + 2nπ/3 and t = −π/2 + 2nπ/3 for n = 1, 2, 3, . . . . 149 3.8 Linear Models: Initial-Value Problems 13. From k1 = 40 and k2 = 120 we compute the effective spring constant k = 4(40)(120)/160 = 120. Now, m = 20/32 so k/m = 120(32)/20 = 192 and x′′ + 192x = 0. Using x(0) = 0 and x′ (0) = 2 we obtain √ √ x(t) = 123 sin 8 3 t. 14. Let m be the mass and k1 and k2 the spring constants. Then k = 4k1 k2 /(k1 + k2 ) is the effective spring constant of the system. Since the initial mass stretches one spring 13 foot and another spring 12 foot, using F = ks, we have 31 k1 = 12 k2 or 2k1 = 3k2 . The given period of the combined system is 2π/ω = π/15, so ω = 30. Since a mass weighing 8 pounds is 41 slug, we have from w2 = k/m 302 = k = 4k 1/4 or k = 225. We now have the system of equations 4k1 k2 = 225 k1 + k2 2k1 = 3k2 . Solving the second equation for k1 and substituting in the first equation, we obtain 12k22 12k2 4(3k2 /2)k2 = = = 225. 3k2 /2 + k2 5k2 5 Thus, k2 = 375/4 and k1 = 1125/8. Finally, the weight of the first mass is 32m = k1 1125/8 375 = = ≈ 46.88 lb. 3 3 8 15. For large values of t the differential equation is approximated by x′′ = 0. The solution of this equation is the linear function x = c1 t + c2 . Thus, for large time, the restoring force will have decayed to the point where the spring is incapable of returning the mass, and the spring will simply keep on stretching. 16. As t becomes larger the spring constant increases; that is, the spring is stiffening. It would seem that the oscillations would become periodic and the spring would oscillate more rapidly. It is likely that the amplitudes of the oscillations would decrease as t increases. 17. (a) above (b) heading upward 18. (a) below (b) from rest 19. (a) below (b) heading upward 20. (a) above (b) heading downward 1 ′′ 8x + x′ + 2x = 0, x(0) = −1, and x′ (0) = 8 we obtain x = 4te−4t − e−4t and x′ = 8e−4t − 16te−4t . If x = 0 then t = 1/4 second. If x′ = 0 then t = 1/2 second and the extreme displacement is x = e−2 feet. √ √ √ " √ ! 22. From 41 x′′ + 2 x′ + 2x = 0, x(0) = 0, and x′ (0) = 5 we obtain x = 5te−2 2 t and x′ = 5e−2 2 t 1 − 2 2 t . If √ √ x′ = 0 then t = 2/4 second and the extreme displacement is x = 5 2 e−1 /4 feet. 21. From 23. (a) From x′′ + 10x′ + 16x = 0, x(0) = 1, and x′ (0) = 0 we obtain x = 43 e−2t − 31 e−8t . (b) From x′′ + x′ + 16x = 0, x(0) = 1, and x′ (0) = −12 then x = − 23 e−2t + 35 e−8t . ! " 24. (a) x = 13 e−8t 4e6t − 1 is not zero for t ≥ 0; the extreme displacement is x(0) = 1 meter. ! " ! " (b) x = 13 e−8t 5 − 2e6t = 0 when t = 61 ln 25 ≈ 0.153 second; if x′ = 43 e−8t e6t − 10 = 0 then t = 0.384 second and the extreme displacement is x = −0.232 meter. 150 1 6 ln 10 ≈ 3.8 Linear Models: Initial-Value Problems 25. (a) From 0.1x′′ + 0.4x′ + 2x = 0, x(0) = −1, and x′ (0) = 0 we obtain x = e−2t − cos 4t − √ 5 −2t e sin(4t + 4.25) (b) x = 2 1 2 . sin 4t . (c) If x = 0 then 4t + 4.25 = 2π, 3π, 4π, . . . so that the first time heading upward is t = 1.294 seconds. ! " 26. (a) From 14 x′′ + x′ + 5x = 0, x(0) = 1/2, and x′ (0) = 1 we obtain x = e−2t 21 cos 4t + 12 sin 4t . # 1 π$ . (b) x = √ e−2t sin 4t + 4 2 (c) If x = 0 then 4t + π/4 = π, 2π, 3π, . . . so that the times heading downward are t = (7 + 8n)π/16 for n = 0, 1, 2, . . . . (d) x 1 .5 .5 2 t -.5 -1 27. From 5 ′′ 16 x + βx′ + 5x = 0 we find that the roots of the auxiliary equation are m = − 85 β ± (a) If 4β 2 − 25 > 0 then β > 5/2. 4 5 3 4β 2 − 25 . (b) If 4β 2 − 25 = 0 then β = 5/2. (c) If 4β 2 − 25 < 0 then 0 < β < 5/2. √ 28. From 0.75x′′ + βx′ + 6x = 0 and β > 3 2 we find that the roots of the auxiliary equation are 3 m = − 32 β ± 23 β 2 − 18 and + * 23 2 23 2 −2βt/3 β − 18 t + c2 sinh β − 18 t . x=e c1 cosh 3 3 3 If x(0) = 0 and x′ (0) = −2 then c1 = 0 and c2 = −3/ β 2 − 18. 29. If 21 x′′ + 12 x′ + 6x = 10 cos 3t, x(0) = −2, and x′ (0) = 0 then / 0 √ √ 47 47 xc = e−t/2 c1 cos t + c2 sin t 2 2 and xp = 10 3 (cos 3t + sin 3t) so that the equation of motion is / 0 √ √ 4 47 47 64 10 −t/2 − cos x=e t − √ sin t + (cos 3t + sin 3t). 3 2 2 3 3 47 30. (a) If x′′ + 2x′ + 5x = 12 cos 2t + 3 sin 2t, x(0) = 1, and x′ (0) = 5 then xc = e−t (c1 cos 2t + c2 sin 2t) and xp = 3 sin 2t so that the equation of motion is x = e−t cos 2t + 3 sin 2t. 151 3.8 Linear Models: Initial-Value Problems x (b) (c) steady-state 3 2 -3 4 6 x 3 x=xc+xp t 2 4 6 t -3 transient 31. From x′′ + 8x′ + 16x = 8 sin 4t, x(0) = 0, and x′ (0) = 0 we obtain xc = c1 e−4t + c2 te−4t and xp = − 41 cos 4t so that the equation of motion is 1 1 x = e−4t + te−4t − cos 4t. 4 4 32. From x′′ + 8x′ + 16x = e−t sin 4t, x(0) = 0, and x′ (0) = 0 we obtain xc = c1 e−4t + c2 te−4t and xp = 24 −t − 625 e cos 4t − 7 −t 625 e sin 4t so that x= 1 −t 1 −4t e (24 + 100t) − e (24 cos 4t + 7 sin 4t). 625 625 As t → ∞ the displacement x → 0. 33. From 2x′′ + 32x = 68e−2t cos 4t, x(0) = 0, and x′ (0) = 0 we obtain xc = c1 cos 4t + c2 sin 4t and xp = 1 −2t 2e cos 4t − 2e−2t sin 4t so that 1 9 1 x = − cos 4t + sin 4t + e−2t cos 4t − 2e−2t sin 4t. 2 4 2 34. Since x = √ 85 4 sin(4t − 0.219) − √ 17 −2t 2 e sin(4t − 2.897), the amplitude approaches √ 85/4 as t → ∞. 35. (a) By Hooke’s law the external force is F (t) = kh(t) so that mx′′ + βx′ + kx = kh(t). (b) From 12 x′′ + 2x′ + 4x = 20 cos t, x(0) = 0, and x′ (0) = 0 we obtain xc = e−2t (c1 cos 2t + c2 sin 2t) and 32 xp = 56 13 cos t + 13 sin t so that ) ( 72 56 32 56 −2t sin 2t + cos t + sin t. x=e − cos 2t − 13 13 13 13 36. (a) From 100x′′ + 1600x = 1600 sin 8t, x(0) = 0, and x′ (0) = 0 we obtain xc = c1 cos 4t + c2 sin 4t and xp = − 13 sin 8t so that by a trig identity x= (b) If x = 1 3 2 1 2 2 sin 4t − sin 8t = sin 4t − sin 4t cos 4t. 3 3 3 3 sin 4t(2 − 2 cos 4t) = 0 then t = nπ/4 for n = 0, 1, 2, . . . . 8 3 cos 4t − 38 cos 8t = 38 (1 − cos 4t)(1 + 2 cos 4t) = 0 then t = π/3 + nπ/2 and t = π/6 + nπ/2 for n = 0, 1, 2, . . . at the extreme values. Note: There are many other values of t for which x′ = 0. √ √ (d) x(π/6 + nπ/2) = 3/2 cm and x(π/3 + nπ/2) = − 3/2 cm. (c) If x′ = (e) x 1 1 2 3 t -1 152 3.8 Linear Models: Initial-Value Problems 37. From x′′ + 4x = −5 sin 2t + 3 cos 2t, x(0) = −1, and x′ (0) = 1 we obtain xc = c1 cos 2t + c2 sin 2t, xp = 3 4 t sin 2t + 54 t cos 2t, and x = − cos 2t − 3 5 1 sin 2t + t sin 2t + t cos 2t. 8 4 4 38. From x′′ + 9x = 5 sin 3t, x(0) = 2, and x′ (0) = 0 we obtain xc = c1 cos 3t + c2 sin 3t, xp = − 56 t cos 3t, and x = 2 cos 3t + 5 5 sin 3t − t cos 3t. 18 6 39. (a) From x′′ + ω 2 x = F0 cos γt, x(0) = 0, and x′ (0) = 0 we obtain xc = c1 cos ωt + c2 sin ωt and xp = " ! (F0 cos γt)/ ω 2 − γ 2 so that x=− (b) lim γ→ω ω2 F0 F0 cos ωt + 2 cos γt. 2 −γ ω − γ2 −F0 t sin γt F0 F0 (cos γt − cos ωt) = lim = t sin ωt. γ→ω ω2 − γ 2 −2γ 2ω 40. From x′′ + ω 2 x = F0 cos ωt, x(0) = 0, and x′ (0) = 0 we obtain xc = c1 cos ωt + c2 sin ωt and xp = (F0 t/2ω) sin ωt so that x = (F0 t/2ω) sin ωt. 41. (a) From cos(u − v) = cos u cos v + sin u sin v and cos(u + v) = cos u cos v − sin u sin v we obtain sin u sin v = 1 2 [cos(u − v) − cos(u + v)]. Letting u = 21 (γ − ω)t and v = 12 (γ + ω)t, the result follows. (b) If ϵ = 21 (γ − ω) then γ ≈ ω so that x = (F0 /2ϵγ) sin ϵt sin γt. 42. See the article “Distinguished Oscillations of a Forced Harmonic Oscillator” by T.G. Procter in The College Mathematics Journal, March, 1995. In this article the author illustrates that for F0 = 1, λ = 0.01, γ = 22/9, and ω = 2 the system exhibits beats oscillations on the interval [0, 9π], but that this phenomenon is transient as t → ∞. x 1 t π −1 9π 43. (a) The general solution of the homogeneous equation is 3 3 xc (t) = c1 e−λt cos( ω 2 − λ2 t) + c2 e−λt sin( ω 2 − λ2 t) 3 = Ae−λt sin[ ω 2 − λ2 t + φ], 3 where A = c21 + c22 , sin φ = c1 /A, and cos φ = c2 /A. Now xp (t) = (ω 2 F0 (ω 2 − γ 2 ) F0 (−2λγ) sin γt + 2 cos γt = A sin(γt + θ), − γ 2 )2 + 4λ2 γ 2 (ω − γ 2 )2 + 4λ2 γ 2 where sin θ = and (ω 2 F0 (−2λγ) − γ 2 )2 + 4λ2 γ 2 F0 3 ω 2 − γ 2 + 4λ2 γ 2 =3 153 (ω 2 −2λγ − γ 2 )2 + 4λ2 γ 2 3.8 Linear Models: Initial-Value Problems cos θ = (ω 2 F0 (ω 2 − γ 2 ) − γ 2 )2 + 4λ2 γ 2 =3 F0 3 (ω 2 − γ 2 )2 + 4λ2 γ 2 (ω 2 ω2 − γ 2 − γ 2 )2 + 4λ2 γ 2 . √ " ! (b) If g ′ (γ) = 0 then γ γ 2 + 2λ2 − ω 2 = 0 so that γ = 0 or γ = ω 2 − 2λ2 . The first derivative test shows √ that g has a maximum value at γ = ω 2 − 2λ2 . The maximum value of g is #3 $ 3 g ω 2 − 2λ2 = F0 /2λ ω 2 − λ2 . 3 √ (c) We identify ω 2 = k/m = 4, λ = β/2, and γ1 = ω 2 − 2λ2 = 4 − β 2 /2 . As β → 0, γ1 → 2 and the resonance curve grows without bound at γ1 = 2. That is, the system approaches pure resonance. g β 2.00 1.00 0.75 0.50 0.25 γ1 1.41 1.87 1.93 1.97 1.99 g 0.58 1.03 1.36 2.02 4.01 4 β=1/4 β=0 3 2 1 1 44. (a) For n = 2, sin2 γt = γ1 = ω/2. 1 2 (1 2 β=1/2 β=0 β=3/4 β=0 β=1 β=1 β=2 β=2 γ 3 4 − cos 2γt). The system is in pure resonance when 2γ1 /2π = ω/2π, or when (b) Note that sin3 γt = sin γt sin2 γt = 1 [sin γt − sin γt cos 2γt]. 2 Now sin(A + B) + sin(A − B) = 2 sin A cos B so sin γt cos 2γt = 1 [sin 3γt − sin γt] 2 and sin3 γt = 1 3 sin γt − sin 3γt. 4 4 Thus x′′ + ω 2 x = 3 1 sin γt − sin 3γt. 4 4 The frequency of free vibration is ω/2π. Thus, when γ1 /2π = ω/2π or γ1 = ω, and when 3γ2 /2π = ω/2π or 3γ2 = ω or γ3 = ω/3, the system will be in pure resonance. (c) γ1=1/2 x 10 10 n=2 -10 x n=3 10 20 30 t -5 γ2=1/3 10 5 5 -5 γ1=1 x n=3 5 10 -10 20 30 t -5 -10 154 20 40 t 3.8 45. Solving 1 ′′ 20 q Linear Models: Initial-Value Problems + 2q ′ + 100q = 0 we obtain q(t) = e−20t (c1 cos 40t + c2 sin 40t). The initial conditions q(0) = 5 and q ′ (0) = 0 imply c1 = 5 and c2 = 5/2. Thus ) ( 3 5 −20t q(t) = e 5 cos 40t + sin 40t = 25 + 25/4 e−20t sin(40t + 1.1071) 2 and q(0.01) ≈ 4.5676 coulombs. 0.0509 second. The charge is zero for the first time when 40t + 1.1071 = π or t ≈ 46. Solving 41 q ′′ + 20q ′ + 300q = 0 we obtain q(t) = c1 e−20t + c2 e−60t . The initial conditions q(0) = 4 and q ′ (0) = 0 imply c1 = 6 and c2 = −2. Thus q(t) = 6e−20t − 2e−60t . Setting q = 0 we find e40t = 1/3 which implies t < 0. Therefore the charge is not 0 for t ≥ 0. 47. Solving 53 q ′′ + 10q ′ + 30q = 300 we obtain q(t) = e−3t (c1 cos 3t + c2 sin 3t) + 10. The initial conditions q(0) = q ′ (0) = 0 imply c1 = c2 = −10. Thus q(t) = 10 − 10e−3t (cos 3t + sin 3t) and i(t) = 60e−3t sin 3t. Solving i(t) = 0 we see that the maximum charge occurs when t = π/3 and q(π/3) ≈ 10.432. 48. Solving q ′′ + 100q ′ + 2500q = 30 we obtain q(t) = c1 e−50t + c2 te−50t + 0.012. The initial conditions q(0) = 0 and q ′ (0) = 2 imply c1 = −0.012 and c2 = 1.4. Thus, using i(t) = q ′ (t) we get q(t) = −0.012e−50t + 1.4te−50t + 0.012 and i(t) = 2e−50t − 70te−50t . Solving i(t) = 0 we see that the maximum charge occurs when t = 1/35 second and q(1/35) ≈ 0.01871 coulomb. √ √ " ! 49. Solving q ′′ + 2q ′ + 4q = 0 we obtain qc = e−t cos 3 t + sin 3 t . The steady-state charge has the form qp = A cos t + B sin t. Substituting into the differential equation we find (3A + 2B) cos t + (3B − 2A) sin t = 50 cos t. Thus, A = 150/13 and B = 100/13. The steady-state charge is qp (t) = 150 100 cos t + sin t 13 13 and the steady-state current is ip (t) = − 50. From and Z = √ E0 ip (t) = Z 100 150 sin t + cos t. 13 13 ( ) X R sin γt − cos γt Z Z X 2 + R2 we see that the amplitude of ip (t) is 7 E02 X 2 E0 3 2 E02 R2 E0 + = . R + X2 = A= 4 4 2 Z Z Z Z 51. The differential equation is 21 q ′′ +20q ′ +1000q = 100 sin 60t. To use Example 10 in the text we identify E0 = 100 and γ = 60. Then 1 1 1 = (60) − ≈ 13.3333, cγ 2 0.001(60) 3 3 Z = X 2 + R2 = X 2 + 400 ≈ 24.0370, X = Lγ − and 155 3.8 Linear Models: Initial-Value Problems 100 E0 = ≈ 4.1603. Z Z From Problem 50, then ip (t) ≈ 4.1603 sin(60t + φ) where sin φ = −X/Z and cos φ = R/Z. Thus tan φ = −X/R ≈ −0.6667 and φ is a fourth quadrant angle. Now φ ≈ −0.5880 and ip (t) = 4.1603 sin(60t − 0.5880). 52. Solving 21 q ′′ + 20q ′ + 1000q = 0 we obtain qc (t) = e−20t (c1 cos 40t + c2 sin 40t). The steady-state charge has the form qp (t) = A sin 60t + B cos 60t + C sin 40t + D cos 40t. Substituting into the differential equation we find (−1600A − 2400B) sin 60t + (2400A − 1600B) cos 60t + (400C − 1600D) sin 40t + (1600C + 400D) cos 40t = 200 sin 60t + 400 cos 40t. Equating coefficients we obtain A = −1/26, B = −3/52, C = 4/17, and D = 1/17. The steady-state charge is qp (t) = − 1 3 4 1 sin 60t − cos 60t + sin 40t + cos 40t 26 52 17 17 and the steady-state current is ip (t) = − 30 45 160 40 cos 60t + sin 60t + cos 40t − sin 40t. 13 13 17 17 53. Solving 12 q ′′ + 10q ′ + 100q = 150 we obtain q(t) = e−10t (c1 cos 10t + c2 sin 10t) + 3/2. The initial conditions q(0) = 1 and q ′ (0) = 0 imply c1 = c2 = −1/2. Thus 1 3 q(t) = − e−10t (cos 10t + sin 10t) + . 2 2 As t → ∞, q(t) → 3/2. 54. In Problem 50 it is shown that the amplitude of the steady-state current is E0 /Z, where √ Z = X 2 + R2 and X = Lγ − 1/Cγ. Since E0 is constant the amplitude will be a maximum when Z is a minimum. Since R is constant, Z will be a minimum when X = 0. Solving Lγ − 1/Cγ = 0 for γ we obtain √ γ = 1/ LC . The maximum amplitude will be E0 /R. √ 55. By Problem 50 the amplitude of the steady-state current is E0 /Z, where Z = X 2 + R2 and X = Lγ − 1/Cγ. Since E0 is constant the amplitude will be a maximum when Z is a minimum. Since R is constant, Z will be a minimum when X = 0. Solving Lγ − 1/Cγ = 0 for C we obtain C = 1/Lγ 2 . 56. Solving 0.1q ′′ + 10q = 100 sin γt we obtain q(t) = c1 cos 10t + c2 sin 10t + qp (t) where qp (t) = A sin γt + B cos γt. Substituting qp (t) into the differential equation we find (100 − γ 2 )A sin γt + (100 − γ 2 )B cos γt = 100 sin γt. Equating coefficients we obtain A = 100/(100 − γ 2 ) and B = 0. Thus, qp (t) = conditions q(0) = q ′ (0) = 0 imply c1 = 0 and c2 = −10γ/(100 − γ 2 ). The charge is q(t) = 10 (10 sin γt − γ sin 10t) 100 − γ 2 156 100 sin γt. The initial 100 − γ 2 3.9 and the current is i(t) = Linear Models: Boundary-Value Problems 100γ (cos γt − cos 10t). 100 − γ 2 57. In an LC-series circuit there is no resistor, so the differential equation is d2 q 1 + q = E(t). dt2 C $ # √ $ # √ Then q(t) = c1 cos t/ LC + c2 sin t/ LC + qp (t) where qp (t) = A sin γt + B cos γt. Substituting qp (t) into L the differential equation we find ( ) ( ) 1 1 2 2 − Lγ A sin γt + − Lγ B cos γt = E0 cos γt. C C Equating coefficients we obtain A = 0 and B = E0 C/(1 − LCγ 2 ). Thus, the charge is q(t) = c1 cos √ 1 E0 C 1 cos γt. t + c2 sin √ t+ 1 − LCγ 2 LC LC √ The initial conditions q(0) = q0 and q ′ (0) = i0 imply c1 = q0 − E0 C/(1 − LCγ 2 ) and c2 = i0 LC . The current is i(t) = q ′ (t) or c1 1 c2 1 E0 Cγ i(t) = − √ sin γt sin √ t+ √ cos √ t− 1 − LCγ 2 LC LC LC LC ( ) E0 C 1 E0 Cγ 1 1 = i0 cos √ q0 − t− √ t− sin √ sin γt. 2 1 − LCγ 1 − LCγ 2 LC LC LC √ 58. When the circuit is in resonance the form of qp (t) is qp (t) = At cos kt+Bt sin kt where k = 1/ LC . Substituting qp (t) into the differential equation we find qp′′ + k 2 qp = −2kA sin kt + 2kB cos kt = E0 cos kt. L Equating coefficients we obtain A = 0 and B = E0 /2kL. The charge is q(t) = c1 cos kt + c2 sin kt + E0 t sin kt. 2kL The initial conditions q(0) = q0 and q ′ (0) = i0 imply c1 = q0 and c2 = i0 /k. The current is i(t) = −c1 k sin kt + c2 k cos kt + = ( E0 (kt cos kt + sin kt) 2kL ) E0 E0 − q0 k sin kt + i0 cos kt + t cos kt. 2kL 2L EXERCISES 3.9 Linear Models: Boundary-Value Problems 1. (a) The general solution is y(x) = c1 + c2 x + c3 x2 + c4 x3 + 157 w0 4 x . 24EI 3.9 Linear Models: Boundary-Value Problems The boundary conditions are y(0) = 0, y ′ (0) = 0, y ′′ (L) = 0, y ′′′ (L) = 0. The first two conditions give c1 = 0 and c2 = 0. The conditions at x = L give the system w0 2 2c3 + 6c4 L + L =0 2EI w0 6c4 + L = 0. EI Solving, we obtain c3 = w0 L2 /4EI and c4 = −w0 L/6EI. The deflection is w0 y(x) = (6L2 x2 − 4Lx3 + x4 ). 24EI x (b) 0.2 0.4 0.6 0.8 1 1 2 3 y 2. (a) The general solution is w0 4 x . 24EI The boundary conditions are y(0) = 0, y ′′ (0) = 0, y(L) = 0, y ′′ (L) = 0. The first two conditions give c1 = 0 y(x) = c1 + c2 x + c3 x2 + c4 x3 + and c3 = 0. The conditions at x = L give the system w0 4 L =0 c2 L + c4 L3 + 24EI w0 2 6c4 L + L = 0. 2EI Solving, we obtain c2 = w0 L3 /24EI and c4 = −w0 L/12EI. The deflection is w0 (L3 x − 2Lx3 + x4 ). y(x) = 24EI (b) 0.2 0.4 0.6 0.8 1 x 1 y 3. (a) The general solution is w0 4 x . 24EI The boundary conditions are y(0) = 0, y ′ (0) = 0, y(L) = 0, y ′′ (L) = 0. The first two conditions give c1 = 0 and c2 = 0. The conditions at x = L give the system w0 4 c3 L2 + c4 L3 + L =0 24EI w0 2 2c3 + 6c4 L + L = 0. 2EI y(x) = c1 + c2 x + c3 x2 + c4 x3 + 158 3.9 Linear Models: Boundary-Value Problems Solving, we obtain c3 = w0 L2 /16EI and c4 = −5w0 L/48EI. The deflection is w0 y(x) = (3L2 x2 − 5Lx3 + 2x4 ). 48EI (b) 0.2 0.4 0.6 0.8 1 1 x y 4. (a) The general solution is w0 L4 π sin x. 4 EIπ L The boundary conditions are y(0) = 0, y ′ (0) = 0, y(L) = 0, y ′′ (L) = 0. The first two conditions give c1 = 0 y(x) = c1 + c2 x + c3 x2 + c4 x3 + and c2 = −w0 L3 /EIπ 3 . The conditions at x = L give the system w0 4 c3 L2 + c4 L3 + L =0 EIπ 3 2c3 + 6c4 L = 0. Solving, we obtain c3 = 3w0 L2 /2EIπ 3 and c4 = −w0 L/2EIπ 3 . The deflection is ) ( w0 L 2L3 π 2 2 3 y(x) = sin x . −2L x + 3Lx − x + 2EIπ 3 π L (b) 0.2 0.4 0.6 0.8 1 x 1 y (c) Using a CAS we find the maximum deflection to be 0.270806 when x = 0.572536. 5. (a) The general solution is w0 x5 . 120EI The boundary conditions are y(0) = 0, y ′′ (0) = 0, y(L) = 0, y ′′ (L) = 0. The first two conditions give c1 = 0 and c3 = 0. The conditions at x = L give the system w0 L5 = 0 c2 L + c4 L3 + 120EI w0 3 6c4 L + L = 0. 6EI Solving, we obtain c2 = 7w0 L4 /360EI and c4 = −w0 L2 /36EI. The deflection is w0 y(x) = (7L4 x − 10L2 x3 + 3x5 ). 360EI y(x) = c1 + c2 x + c3 x2 + c4 x3 + 159 3.9 Linear Models: Boundary-Value Problems (b) 1 x 0.2 0.4 0.6 0.8 1 y (c) Using a CAS we find the maximum deflection to be 0.234799 when x = 0.51933. 6. (a) ymax = y(L) = w0 L4 /8EI (b) Replacing both L and x by L/2 in y(x) we obtain w0 L4 /128EI, which is 1/16 of the maximum deflection when the length of the beam is L. (c) ymax = y(L/2) = 5w0 L4 /384EI (d) The maximum deflection in Example 1 is y(L/2) = (w0 /24EI)L4 /16 = w0 L4 /384EI, which is 1/5 of the maximum displacement of the beam in part c. 7. The general solution of the differential equation is 7 7 w0 2 w0 EI P P . x + c2 sinh x+ x + y = c1 cosh EI EI 2P P2 Setting y(0) = 0 we obtain c1 = −w0 EI/P 2 , so that 7 7 w0 EI w0 2 w0 EI P P cosh x + c2 sinh x+ x + . y=− 2 P EI EI 2P P2 Setting y ′ (L) = 0 we find c2 = /7 P w0 EI sinh EI P 2 7 w0 L P L− EI P 087 P cosh EI 7 P L. EI 8. The general solution of the differential equation is 7 7 w0 2 w0 EI P P y = c1 cos x + c2 sin x+ x + . EI EI 2P P2 Setting y(0) = 0 we obtain c1 = −w0 EI/P 2 , so that 7 7 w0 EI w0 2 w0 EI P P y=− cos . x + c2 sin x+ x + P2 EI EI 2P P2 Setting y ′ (L) = 0 we find c2 = / 7 − P w0 EI sin EI P 2 7 P w0 L L− EI P 087 P cos EI 7 P L. EI 9. This is Example 2 in the text with L = π. The eigenvalues are λn = n2 π 2 /π 2 = n2 , n = 1, 2, 3, . . . and the corresponding eigenfunctions are yn = sin(nπx/π) = sin nx, n = 1, 2, 3, . . . . 10. This is Example 2 in the text with L = π/4. The eigenvalues are λn = n2 π 2 /(π/4)2 = 16n2 , n = 1, 2, 3, . . . and the eigenfunctions are yn = sin(nπx/(π/4)) = sin 4nx, n = 1, 2, 3, . . . . 160 3.9 Linear Models: Boundary-Value Problems 11. For λ ≤ 0 the only solution of the boundary-value problem is y = 0. For λ = α2 > 0 we have y = c1 cos αx + c2 sin αx. Now y ′ (x) = −c1 α sin αx + c2 α cos αx and y ′ (0) = 0 implies c2 = 0, so y(L) = c1 cos αL = 0 gives (2n − 1)2 π 2 , n = 1, 2, 3, . . . . 4L2 (2n − 1)π x for n = 1, 2, 3, . . . . The eigenvalues (2n − 1)2 π 2 /4L2 correspond to the eigenfunctions cos 2L 12. For λ ≤ 0 the only solution of the boundary-value problem is y = 0. For λ = α2 > 0 we have αL = (2n − 1)π 2 or λ = α2 = y = c1 cos αx + c2 sin αx. Since y(0) = 0 implies c1 = 0, y = c2 sin x dx. Now #π$ π y′ = c2 α cos α = 0 2 2 gives π (2n − 1)π α = or λ = α2 = (2n − 1)2 , n = 1, 2, 3, . . . . 2 2 The eigenvalues λn = (2n − 1)2 correspond to the eigenfunctions yn = sin(2n − 1)x. 13. For λ = −α2 < 0 the only solution of the boundary-value problem is y = 0. For λ = 0 we have y = c1 x + c2 . Now y ′ = c1 and y ′ (0) = 0 implies c1 = 0. Then y = c2 and y ′ (π) = 0. Thus, λ = 0 is an eigenvalue with corresponding eigenfunction y = 1. For λ = α2 > 0 we have y = c1 cos αx + c2 sin αx. Now y ′ (x) = −c1 α sin αx + c2 α cos αx and y ′ (0) = 0 implies c2 = 0, so y ′ (π) = −c1 α sin απ = 0 gives απ = nπ or λ = α2 = n2 , n = 1, 2, 3, . . . . The eigenvalues n2 correspond to the eigenfunctions cos nx for n = 0, 1, 2, . . . . 14. For λ ≤ 0 the only solution of the boundary-value problem is y = 0. For λ = α2 > 0 we have y = c1 cos αx + c2 sin αx. Now y(−π) = y(π) = 0 implies c1 cos απ − c2 sin απ = 0 c1 cos απ + c2 sin απ = 0. This homogeneous system will have a nontrivial solution when % % % cos απ − sin απ % % % = 2 sin απ cos απ = sin 2απ = 0. % cos απ sin απ % 161 (1) 3.9 Linear Models: Boundary-Value Problems Then n2 ; n = 1, 2, 3, . . . . 4 When n = 2k − 1 is odd, the eigenvalues are (2k − 1)2 /4. Since cos(2k − 1)π/2 = 0 and sin(2k − 1)π/2 ̸= 0, we see from either equation in (1) that c2 = 0. Thus, the eigenfunctions corresponding to the eigenvalues 2απ = nπ or λ = α2 = (2k − 1)2 /4 are y = cos(2k − 1)x/2 for k = 1, 2, 3, . . . . Similarly, when n = 2k is even, the eigenvalues are k 2 with corresponding eigenfunctions y = sin kx for k = 1, 2, 3, . . . . 15. The auxiliary equation has solutions $ 3 1# −2 ± 4 − 4(λ + 1) = −1 ± α. 2 m= For λ = −α2 < 0 we have y = e−x (c1 cosh αx + c2 sinh αx) . The boundary conditions imply y(0) = c1 = 0 y(5) = c2 e−5 sinh 5α = 0 so c1 = c2 = 0 and the only solution of the boundary-value problem is y = 0. For λ = 0 we have y = c1 e−x + c2 xe−x and the only solution of the boundary-value problem is y = 0. For λ = α2 > 0 we have y = e−x (c1 cos αx + c2 sin αx) . Now y(0) = 0 implies c1 = 0, so y(5) = c2 e−5 sin 5α = 0 gives n2 π 2 , n = 1, 2, 3, . . . . 25 nπ n2 π 2 correspond to the eigenfunctions yn = e−x sin x for n = 1, 2, 3, . . . . The eigenvalues λn = 25 5 16. For λ < −1 the only solution of the boundary-value problem is y = 0. For λ = −1 we have y = c1 x + c2 . Now y ′ = c1 and y ′ (0) = 0 implies c1 = 0. Then y = c2 and y ′ (1) = 0. Thus, λ = −1 is an eigenvalue with 5α = nπ or λ = α2 = corresponding eigenfunction y = 1. For λ > −1 or λ + 1 = α2 > 0 we have y = c1 cos αx + c2 sin αx. Now y ′ = −c1 α sin αx + c2 α cos αx and y ′ (0) = 0 implies c2 = 0, so y ′ (1) = −c1 α sin α = 0 gives α = nπ, λ + 1 = α 2 = n2 π 2 , or λ = n2 π 2 − 1, n = 1, 2, 3, . . . . The eigenvalues n2 π 2 − 1 correspond to the eigenfunctions cos nπx for n = 0, 1, 2, . . . . 162 3.9 Linear Models: Boundary-Value Problems 17. For λ = α2 > 0 a general solution of the given differential equation is y = c1 cos(α ln x) + c2 sin(α ln x). Since ln 1 = 0, the boundary condition y(1) = 0 implies c1 = 0. Therefore y = c2 sin(α ln x). Using ln eπ = π we find that y (eπ ) = 0 implies c2 sin απ = 0 or απ = nπ, n = 1, 2, 3, . . . . The eigenvalues and eigenfunctions are, in turn, λ = α 2 = n2 , n = 1, 2, 3, . . . and y = sin(n ln x). For λ ≤ 0 the only solution of the boundary-value problem is y = 0. 18. For λ = 0 the general solution is y = c1 + c2 ln x. Now y ′ = c2 /x, so y ′ (e−1 ) = c2 e = 0 implies c2 = 0. Then y = c1 and y(1) = 0 gives c1 = 0. Thus y(x) = 0. For λ = −α2 < 0, y = c1 x−α + c2 xα . The boundary conditions give c2 = c1 e2α and c1 = 0, so that c2 = 0 and y(x) = 0. For λ = α2 > 0, y = c1 cos(α ln x) + c2 sin(α ln x). From y(1) = 0 we obtain c1 = 0 and y = c2 sin(α ln x). Now y ′ = c2 (α/x) cos(α ln x), so y ′ (e−1 ) = c2 eα cos α = 0 implies cos α = 0 or α = (2n − 1)π/2 and λ = α2 = (2n − 1)2 π 2 /4 for n = 1, 2, 3, . . . . The corresponding eigenfunctions are ) ( 2n − 1 yn = sin π ln x . 2 19. For λ = α4 , α > 0, the general solution of the boundary-value problem y (4) − λy = 0, y(0) = 0, y ′′ (0) = 0, y(1) = 0, y ′′ (1) = 0 is y = c1 cos αx + c2 sin αx + c3 cosh αx + c4 sinh αx. The boundary conditions y(0) = 0, y ′′ (0) = 0 give c1 + c3 = 0 and −c1 α2 + c3 α2 = 0, from which we conclude c1 = c3 = 0. Thus, y = c2 sin αx + c4 sinh αx. The boundary conditions y(1) = 0, y ′′ (1) = 0 then give c2 sin α + c4 sinh α = 0 −c2 α2 sin α + c4 α2 sinh α = 0. In order to have nonzero solutions of this system, we must have the determinant of the coefficients equal zero, that is, % % % sin α sinh α % % % or 2α2 sinh α sin α = 0. % −α2 sin α α2 sinh α % = 0 But since α > 0, the only way that this is satisfied is to have sin α = 0 or α = nπ. The system is then satisfied by choosing c2 ̸= 0, c4 = 0, and α = nπ. The eigenvalues and corresponding eigenfunctions are then λn = α4 = (nπ)4 , n = 1, 2, 3, . . . and y = sin nπx. 20. For λ = α4 , α > 0, the general solution of the differential equation is y = c1 cos αx + c2 sin αx + c3 cosh αx + c4 sinh αx. 163 3.9 Linear Models: Boundary-Value Problems The boundary conditions y ′ (0) = 0, y ′′′ (0) = 0 give c2 α + c4 α = 0 and −c2 α3 + c4 α3 = 0 from which we conclude c2 = c4 = 0. Thus, y = c1 cos αx + c3 cosh αx. The boundary conditions y(π) = 0, y ′′ (π) = 0 then give c2 cos απ + c4 cosh απ = 0 2 −c2 λ cos απ + c4 λ2 cosh απ = 0. The determinant of the coefficients is 2α2 cosh α cos α = 0. But since α > 0, the only way that this is satisfied is to have cos απ = 0 or α = (2n − 1)/2, n = 1, 2, 3, . . . . The eigenvalues and corresponding eigenfunctions are ( ( )4 ) 2n − 1 2n − 1 λn = α 4 = , n = 1, 2, 3, . . . and y = cos x. 2 2 y 21. If restraints are put on the column at x = L/4, x = L/2, and x = 3L/4, then the critical load will be P4 . L x 22. (a) The general solution of the differential equation is 7 7 P P x + c2 sin x + δ. y = c1 cos EI EI Since the column is embedded at x = 0, the boundary conditions are y(0) = y ′ (0) = 0. If δ = 0 this implies that c1 = c2 = 0 and y(x) = 0. That is, there is no deflection. (b) If δ ̸= 0, the boundary conditions give, in turn, c1 = −δ and c2 = 0. Then / 0 7 P y = δ 1 − cos x . EI In order to satisfy the boundary condition y(L) = δ we must have 0 / 7 7 P P L or cos L = 0. δ = δ 1 − cos EI EI This gives 3 P/EI L = nπ/2 for n = 1, 2, 3, . . . . The smallest value of Pn , the Euler load, is then 7 ) ( π 1 π 2 EI P1 L= or P1 = . EI 2 4 L2 23. If λ = α2 = P/EI, then the solution of the differential equation is y = c1 cos αx + c2 sin αx + c3 x + c4 . The conditions y(0) = 0, y ′′ (0) = 0 yield, in turn, c1 + c4 = 0 and c1 = 0. With c1 = 0 and c4 = 0 the solution is y = c2 sin αx + c3 x. The conditions y(L) = 0, y ′′ (L) = 0, then yield c2 sin αL + c3 L = 0 and c2 sin αL = 0. Hence, nontrivial solutions of the problem exist only if sin αL = 0. From this point on, the analysis is the same as in Example 3 in the text. 164 3.9 Linear Models: Boundary-Value Problems 24. (a) The boundary-value problem is d2 y d4 y + λ 2 = 0, 4 dx dx y(0) = 0, y ′′ (0) = 0, y(L) = 0, y ′ (L) = 0, where λ = α2 = P/EI. The solution of the differential equation is y = c1 cos αx + c2 sin αx + c3 x + c4 and the conditions y(0) = 0, y ′′ (0) = 0 yield c1 = 0 and c4 = 0. Next, by applying y(L) = 0, y ′ (L) = 0 to y = c2 sin αx + c3 x we get the system of equations c2 sin αL + c3 L = 0 αc2 cos αL + c3 = 0. To obtain nontrivial solutions c2 , c3 , we must have the determinant of the coefficients equal to zero: % % % sin αL L % % % or tan β = β, % α cos αL 1 % = 0 where β = αL. If βn denotes the positive roots of the last equation, then the eigenvalues are found from √ λn L or λn = (βn /L)2 . From λ = P/EI we see that the critical loads are Pn = βn2 EI/L2 . With the aid of a CAS we find that the first positive root of tan β = β is (approximately) β1 = 4.4934, and βn = αn L = so the Euler load is (approximately) P1 = 20.1907EI/L2 . Finally, if we use c3 = −c2 α cos αL, then the deflection curves are * ( ) ( ) + βn βn yn (x) = c2 sin αn x + c3 x = c2 sin x − cos βn x . L L (b) With L = 1 and c2 appropriately chosen, the general shape of the first buckling mode, is shown below. * ( ) ( ) + 4.4934 4.4934 y1 (x) = c2 sin x − cos(4.4934) x , L L y1 0.2 25. The general solution is 0.4 0.6 0.8 7 1 x 7 ρ ρ y = c1 cos ωx + c2 sin ωx. T T 3 From y(0) = 0 we obtain c1 = 0. Setting y(L) = 0 we find ρ/T ωL = nπ, n = 1, 2, 3, . . . . Thus, critical √ √ speeds are ωn = nπ T /L ρ , n = 1, 2, 3, . . . . The corresponding deflection curves are y(x) = c2 sin nπ x, L n = 1, 2, 3, . . . , where c2 ̸= 0. 26. (a) When T (x) = x2 the given differential equation is the Cauchy-Euler equation x2 y ′′ + 2xy ′ + ρω 2 y = 0. The solutions of the auxiliary equation m(m − 1) + 2m + ρω 2 = m2 + m + ρω 2 = 0 165 3.9 Linear Models: Boundary-Value Problems are 1 13 1 13 m1 = − − 4ρω 2 − 1 i, m2 = − + 4ρω 2 − 1 i 2 2 2 2 when ρω 2 > 0.25. Thus y = c1 x−1/2 cos(λ ln x) + c2 x−1/2 sin(λ ln x) 3 where λ = 21 4ρω 2 − 1. Applying y(1) = 0 gives c1 = 0 and consequently y = c2 x−1/2 sin(λ ln x). The condition y(e) = 0 requires c2 e−1/2 sin λ = 0. We obtain a nontrivial solution when λn = nπ, n = 1, 2, 3, . . . . But λn = Solving for ωn gives ωn = The corresponding solutions are 13 4ρωn2 − 1 = nπ. 2 13 2 2 (4n π + 1)/ρ . 2 yn (x) = c2 x−1/2 sin(nπ ln x). (b) y y 1 y 1 n=1 1 n=3 n=2 ex 1 ex 1 -1 -1 1 ex -1 27. The auxiliary equation is m2 +m = m(m+1) = 0 so that u(r) = c1 r−1 +c2 . The boundary conditions u(a) = u0 and u(b) = u1 yield the system c1 a−1 + c2 = u0 , c1 b−1 + c2 = u1 . Solving gives ( ) u0 − u1 u1 b − u0 a c1 = . ab and c2 = b−a b−a Thus u(r) = ( u0 − u1 b−a ) ab u1 b − u0 a + . r b−a 28. The auxiliary equation is m2 = 0 so that u(r) = c1 + c2 ln r. The boundary conditions u(a) = u0 and u(b) = u1 yield the system c1 + c2 ln a = u0 , c1 + c2 ln b = u1 . Solving gives c1 = Thus u(r) = u1 ln a − u0 ln b ln(a/b) and c2 = u0 − u1 . ln(a/b) u0 ln(r/b) − u1 ln(r/a) u1 ln a − u0 ln b u0 − u1 + ln r = . ln(a/b) ln(a/b) ln(a/b) 29. The solution of the initial-value problem x′′ + ω 2 x = 0, x(0) = 0, x′ (0) = v0 , ω 2 = 10/m is x(t) = (v0 /ω) sin ωt. To satisfy the additional boundary condition x(1) = 0 we require that ω = nπ, n = 1, 2, 3, . . . . The eigenvalues λ = ω 2 = n2 π 2 and eigenfunctions of the problem are then x(t) = (v0 /nπ) sin nπt. Using ω 2 = 10/m we find that the only masses that can pass through the equilibrium position at t = 1 are mn = 10/n2 π 2 . Note for n = 1, the heaviest mass m1 = 10/π 2 will not pass through the 166 3.9 Linear Models: Boundary-Value Problems equilibrium position on the interval 0 < t < 1 (the period of x(t) = (v0 /π) sin πt is T = 2, so on 0 ≤ t ≤ 1 its graph passes through x = 0 only at t = 0 and t = 1). Whereas for n > 1, masses of lighter weight will pass through the equilibrium position n − 1 times prior to passing through at t = 1. For example, if n = 2, the period of x(t) = (v0 /2π) sin 2πt is 2π/2π = 1, the mass will pass through x = 0 only once (t = 12 ) prior to t = 1; if n = 3, the period of x(t) = (v0 /3π) sin 3πt is to t = 1; and so on. 2 3 , the mass will pass through x = 0 twice (t = 1 3 and t = 23 ) prior 30. The initial-value problem is k 2 ′ x + x = 0, m m x(0) = 0, x′ (0) = v0 . √ With k = 10, the auxiliary equation has roots γ = −1/m ± 1 − 10m/m. Consider the three cases: x′′ + 1 . The roots are γ1 = γ2 = 10 and the solution of the differential equation is x(t) = c1 e−10t +c2 te−10t . (i) m = 10 The initial conditions imply c1 = 0 and c2 = v0 and so x(t) = v0 te−10t . The condition x(1) = 0 implies v0 e−10 = 0 which is impossible because v0 ̸= 0. 1 10 (ii) 1 − 10m > 0 or 0 < m < γ1 = − . The roots are 1 1√ 1 − 10m − m m and γ2 = − 1 1√ 1 − 10m + m m and the solution of the differential equation is x(t) = c1 eγ1 t + c2 eγ2 t . The initial conditions imply c1 + c2 = 0 γ1 c1 + γ2 c2 = v0 so c1 = v0 /(γ1 − γ2 ), c2 = −v0 /(γ1 − γ2 ), and x(t) = v0 (eγ1 t − eγ2 t ). γ1 − γ2 Again, x(1) = 0 is impossible because v0 ̸= 0. (iii) 1 − 10m < 0 or m > 1 10 . The roots of the auxiliary equation are γ1 = − 1√ 1 − 10m − 1 i m m and γ2 = − 1√ 1 + 10m − 1 i m m and the solution of the differential equation is 1√ 1√ 10m − 1 t + c2 e−t/m sin 10m − 1 t. m m √ The initial conditions imply c1 = 0 and c2 = mv0 / 10m − 1, so that ( ) mv0 1 √ −t/m sin 10m − 1 t , x(t) = √ e m 10m − 1 x(t) = c1 e−t/m cos The condition x(1) = 0 implies √ mv0 1√ 10m − 1 = 0 e−1/m sin m 10m − 1 1√ sin 10m − 1 = 0 m 1√ 10m − 1 = nπ m 10m − 1 = n2 π 2 , n = 1, 2, 3, . . . m2 (n2 π 2 )m2 − 10m + 1 = 0 167 3.9 Linear Models: Boundary-Value Problems √ √ 10 100 − 4n2 π 2 5 ± 25 − n2 π 2 m= = . 2n2 π 2 n2 π 2 Since m is real, 25 − n2 π 2 ≥ 0. If 25 − n2 π 2 = 0, then n2 = 25/π 2 , and n is not an integer. Thus, 25 − n2 π 2 = (5 − nπ)(5 + nπ) > 0 and since n > 0, 5 + nπ > 0, so 5 − nπ > 0 also. Then n < 5/π, and so n = 1. Therefore, the mass m will pass through the equilibrium position when t = 1 for √ √ 5 + 25 − π 2 5 − 25 − π 2 m1 = and m2 = . π2 π2 31. (a) The general solution of the differential equation is y = c1 cos 4x + c2 sin 4x. From y0 = y(0) = c1 we see that y = y0 cos 4x + c2 sin 4x. From y1 = y(π/2) = y0 we see that any solution must satisfy y0 = y1 . We also see that when y0 = y1 , y = y0 cos 4x + c2 sin 4x is a solution of the boundary-value problem for any choice of c2 . Thus, the boundary-value problem does not have a unique solution for any choice of y0 and y1 . (b) Whenever y0 = y1 there are infinitely many solutions. (c) When y0 ̸= y1 there will be no solutions. (d) The boundary-value problem will have the trivial solution when y0 = y1 = 0. This solution will not be unique. 32. (a) The general solution of the differential equation is y = c1 cos 4x + c2 sin 4x. From 1 = y(0) = c1 we see that y = cos 4x + c2 sin 4x. From 1 = y(L) = cos 4L + c2 sin 4L we see that c2 = (1 − cos 4L)/ sin 4L. Thus, ( ) 1 − cos 4L y = cos 4x + sin 4x sin 4L will be a unique solution when sin 4L ̸= 0; that is, when L ̸= kπ/4 where k = 1, 2, 3, . . . . (b) There will be infinitely many solutions when sin 4L = 0 and 1 − cos 4L = 0; that is, when L = kπ/2 where k = 1, 2, 3, . . . . (c) There will be no solution when sin 4L ̸= 0 and 1 − cos 4L ̸= 0; that is, when L = kπ/4 where k = 1, 3, 5, . . . . (d) There can be no trivial solution since it would fail to satisfy the boundary conditions. 33. (a) A solution curve has the same y-coordinate at both ends of the interval [−π, π] and the tangent lines at the endpoints of the interval are parallel. (b) For λ = 0 the solution of y ′′ = 0 is y = c1 x + c2 . From the first boundary condition we have y(−π) = −c1 π + c2 = y(π) = c1 π + c2 or 2c1 π = 0. Thus, c1 = 0 and y = c2 . This constant solution is seen to satisfy the boundary-value problem. For λ = −α2 < 0 we have y = c1 cosh αx + c2 sinh αx. In this case the first boundary condition gives y(−π) = c1 cosh(−απ) + c2 sinh(−απ) = c1 cosh απ − c2 sinh απ = y(π) = c1 cosh απ + c2 sinh απ or 2c2 sinh απ = 0. Thus c2 = 0 and y = c1 cosh αx. The second boundary condition implies in a similar fashion that c1 = 0. Thus, for λ < 0, the only solution of the boundary-value problem is y = 0. 168 3.9 Linear Models: Boundary-Value Problems For λ = α2 > 0 we have y = c1 cos αx + c2 sin αx. The first boundary condition implies y(−π) = c1 cos(−απ) + c2 sin(−απ) = c1 cos απ − c2 sin απ = y(π) = c1 cos απ + c2 sin απ or 2c2 sin απ = 0. Similarly, the second boundary condition implies 2c1 α sin απ = 0. If c1 = c2 = 0 the solution is y = 0. However, if c1 ̸= 0 or c2 ̸= 0, then sin απ = 0, which implies that α must be an integer, n. Therefore, for c1 and c2 not both 0, y = c1 cos nx + c2 sin nx is a nontrivial solution of the boundary-value problem. Since cos(−nx) = cos nx and sin(−nx) = − sin nx, we may assume without loss of generality that the eigenvalues are λn = α2 = n2 , for n a positive integer. The corresponding eigenfunctions are yn = cos nx and yn = sin nx. y (c) y 3 -p 3 px -p -3 -3 y = 2 sin 3x 2 px √ y = sin 4x − 2 cos 3x √ α x + c2 sin α x. Setting y(0) = 0 we find c1 = 0, so that 34. For λ = α > 0 the general solution is y = c1 cos √ y = c2 sin α x. The boundary condition y(1) + y ′ (1) = 0 implies √ √ √ c2 sin α + c2 α cos α = 0. √ √ Taking c2 ̸= 0, this equation is equivalent to tan α = − α . Thus, the eigenvalues are λn = αn2 = x2n , √ √ n = 1, 2, 3, . . . , where the xn are the consecutive positive roots of tan α = − α . 35. We see from the graph that tan x = −x has infinitely many roots. Since λn = αn2 , there are no new eigenvalues when αn < 0. For λ = 0, the differential equation y ′′ = 0 has general solution y = c1 x + c2 . The boundary conditions imply c1 = c2 = 0, so y = 0. tan x 5 2.5 2 4 6 8 10 12 x -2.5 -5 -7.5 -10 36. Using a CAS we find that the first four nonnegative roots of tan x = −x are approximately 2.02876, 4.91318, 7.97867, and 11.0855. The corresponding eigenvalues are 4.11586, 24.1393, 63.6591, and 122.889, with eigenfunctions sin(2.02876x), sin(4.91318x), sin(7.97867x), and sin(11.0855x). 169 3.9 Linear Models: Boundary-Value Problems 37. In the case when λ = −α2 < 0, the solution of the differential equation is y = c1 cosh αx + c2 sinh αx. The condition y(0) = 0 gives c1 = 0. The condition y(1) − 21 y ′ (1) = 0 applied to y = c2 sinh αx gives c2 (sinh α − 21 α cosh α) = 0 or tanh α = 12 α. As can be seen from y 1 the figure, the graphs of y = tanh x and y = 12 x intersect at a single point with approximate x-coordinate α1 = 1.915. Thus, there is a single negative eigenvalue λ1 = −α12 ≈ −3.667 and the corresponding eigenfuntion is y1 = sinh 1.915x. 1 x 2 For λ = 0 the only solution of the boundary-value problem is y = 0. For λ = α2 > 0 the solution of the differential equation is y = c1 cos αx + c2 sin αx. The condition y(0) = 0 gives c1 = 0, so y = c2 sin αx. The condition y(1) − 12 y ′ (1) = 0 gives c2 (sin α − 21 α cos α) = 0, so the eigenvalues are λn = αn2 when αn , n = 2, 3, 4, . . . , are the positive roots of tan α = 1 2 α. Using a CAS we find that the first three values of α are α2 = 4.27487, α3 = 7.59655, and α4 = 10.8127. The first three eigenvalues are then λ2 = α22 = 18.2738, λ3 = α32 = 57.7075, and λ4 = α42 = 116.9139 with corresponding eigenfunctions y2 = sin 4.27487x, y3 = sin 7.59655x, and y4 = sin 10.8127x. 38. For λ = α4 , α > 0, the solution of the differential equation is y = c1 cos αx + c2 sin αx + c3 cosh αx + c4 sinh αx. y 1 The boundary conditions y(0) = 0, y ′ (0) = 0, y(1) = 0, y ′ (1) = 0 give, in turn, c1 + c3 = 0 2 4 6 8 10 12 x αc2 + αc4 = 0, c1 cos α + c2 sin α + c3 cosh α + c4 sinh α = 0 −c1 α sin α + c2 α cos α + c3 α sinh α + c4 α cosh α = 0. The first two equations enable us to write c1 (cos α − cosh α) + c2 (sin α − sinh α) = 0 c1 (− sin α − sinh α) + c2 (cos α − cosh α) = 0. The determinant % % cos α − cosh α % % − sin α − sinh α % sin α − sinh α % %=0 cos α − cosh α % simplifies to cos α cosh α = 1. From the figure showing the graphs of 1/ cosh x and cos x, we see that this equation has an infinite number of positive roots. With the aid of a CAS the first four roots are found to be α1 = 4.73004, α2 = 7.8532, α3 = 10.9956, and α4 = 14.1372, and the corresponding eigenvalues are λ1 = 500.5636, λ2 = 3803.5281, λ3 = 14,617.5885, and λ4 = 39,944.1890. Using the third equation in the system to eliminate c2 , we find that the eigenfunctions are yn = (− sin αn + sinh αn )(cos αn x − cosh αn x) + (cos αn − cosh αn )(sin αn x − sinh αn x). 170 3.10 Nonlinear Models EXERCISES 3.10 Nonlinear Models x 2 1. The period corresponding to x(0) = 1, x′ (0) = 1 is approximately 5.6. ′ The period corresponding to x(0) = 1/2, x (0) = −1 is approximately 6.2. 1 4 2 6 8 -1 -2 2. The solutions are not periodic. x 10 8 6 4 2 t -2 3. The period corresponding to x(0) = 1, x′ (0) = 1 is approximately 5.8. The second initial-value problem does not have a periodic solution. x 10 8 6 4 2 4 2 6 8 10 -2 4. Both solutions have periods of approximately 6.3. x 3 2 1 -1 -2 -3 171 2 4 6 8 10 t t 3.10 Nonlinear Models 5. From the graph we see that |x1 | ≈ 1.2. x 4 3 2 1 x1=1.2 x1=1.1 t −1 5 6. From the graphs we see that the interval is approximately (−0.8, 1.1). 10 x 3 2 1 5 10 t −1 7. Since xe0.01x = x[1 + 0.01x + for small values of x, a linearization is x 8. 1 (0.01x)2 + · · · ] ≈ x 2! d2 x + x = 0. dt2 3 t 5 10 15 −3 For x(0) = 1 and x′ (0) = 1 the oscillations are symmetric about the line x = 0 with amplitude slightly greater than 1. For x(0) = −2 and x′ (0) = 0.5 the oscillations are symmetric about the line x = −2 with small amplitude. √ For x(0) = 2 and x′ (0) = 1 the oscillations are symmetric about the line x = 0 with amplitude a little greater than 2. For x(0) = 2 and x′ (0) = 0.5 the oscillations are symmetric about the line x = 2 with small amplitude. For x(0) = −2 and x′ (0) = 0 there is no oscillation; the solution is constant. √ For x(0) = − 2 and x′ (0) = −1 the oscillations are symmetric about the line x = 0 with amplitude a little greater than 2. 172 3.10 Nonlinear Models 9. This is a damped hard spring, so x will approach 0 as t x approaches ∞. 2 4 2 6 8 t -2 10. This is a damped soft spring, so we might expect no oscillatory solutions. However, if the initial conditions are sufficiently small the spring can oscillate. x 5 4 3 2 1 2 4 t -1 -2 11. x 15 x k1 = 0.01 10 2 5 1 10 20 t 30 10 -5 -1 -10 -2 -15 -3 x k1 = 1 3 x k1 = 20 3 3 2 2 1 1 5 10 t -1 -2 -2 -3 -3 t k1 = 100 1 -1 20 2 3 t When k1 is very small the effect of the nonlinearity is greatly diminished, and the system is close to pure resonance. 173 3.10 Nonlinear Models 12. (a) x x 40 40 20 20 20 40 60 80 t 100 20 -20 40 60 80 100 t -20 k ! "0.000465 -40 k ! "0.000466 -40 The system appears to be oscillatory for −0.000465 ≤ k1 < 0 and nonoscillatory for k1 ≤ −0.000466. (b) x x 3 3 2 2 1 1 20 40 60 80 100 120 140 t 20 -1 -2 40 60 80 100 120 140 t -1 -2 k ! "0.3493 -3 k ! "0.3494 -3 The system appears to be oscillatory for −0.3493 ≤ k1 < 0 and nonoscillatory for k1 ≤ −0.3494. 13. For λ2 − ω 2 > 0 we choose λ = 2 and ω = 1 with x(0) = 1 and x′ (0) = 2. For λ2 − ω 2 < 0 we choose λ = 1/3 and ω = 1 with x(0) = −2 and x′ (0) = 4. In both cases the motion corresponds to the overdamped θ 3 λ=1/3, ω=1 λ=2, ω=1 and underdamped cases for spring/mass systems. t 5 10 15 −3 14. (a) Setting dy/dt = v, the differential equation in (13) becomes dv/dt = −gR2 /y 2 . But, by the chain rule, dv/dt = (dv/dy)(dy/dt) = v dv/dt, so v dv/dy = −gR2 /y 2 . Separating variables and integrating we obtain v dv = −gR2 dy y2 and 1 2 gR2 v = + c. 2 y Setting v = v0 and y = R we find c = −gR + 12 v02 and R2 − 2gR + v02 . y √ (b) As y → ∞ we assume that v → 0+ . Then v02 = 2gR and v0 = 2gR . v 2 = 2g (c) Using g = 32 ft/s and R = 4000(5280) ft we find 3 v0 = 2(32)(4000)(5280) ≈ 36765.2 ft/s ≈ 25067 mi/hr. (d) v0 = 3 2(0.165)(32)(1080) ≈ 7760 ft/s ≈ 5291 mi/hr 174 3.10 Nonlinear Models 15. (a) Intuitively, one might expect that only half of a 10-pound chain could be lifted by a 5-pound vertical force. 3 √ (b) Since x = 0 when t = 0, and v = dx/dt = 160 − 64x/3 , we have v(0) = 160 ≈ 12.65 ft/s. 3 (c) Since x should always be positive, we solve x(t) = 0, getting t = 0 and t = 23 5/2 ≈ 2.3717. Since the 3 graph of x(t) is a parabola, the maximum value occurs at tm = 43 5/2 . (This can also be obtained by solving x′ (t) = 0.) At this time the height of the chain is x(tm ) ≈ 7.5 ft. This is higher than predicted because of the momentum generated by the force. When the chain is 5 feet high it still has a positive velocity of about 7.3 ft/s, which keeps it going higher for a while. 16. (a) Setting dx/dt = v, the differential equation becomes (L − x)dv/dt − v 2 = Lg. But, by the Chain Rule, dv/dt = (dv/dx)(dx/dt) = v dv/dx, so (L − x)v dv/dx − v 2 = Lg. Separating variables and integrating we obtain v 1 dv = dx v 2 + Lg L−x 1 ln(v 2 + Lg) = − ln(L − x) + ln c, 2 3 √ so v 2 + Lg = c/(L − x). When x = 0, v = 0, and c = L Lg . Solving for v and simplifying we get 3 Lg(2Lx − x2 ) dx = v(x) = . dt L−x and Again, separating variables and integrating we obtain 3 L−x dx = dt and √ 2Lx − x2 √ = t + c1 . Lg Lg(2Lx − x2 ) √ √ Since x(0) = 0, we have c1 = 0 and 2Lx − x2 / Lg = t. Solving for x we get √ 3 dx Lgt 2 2 x(t) = L − L − Lgt and v(t) = . =3 dt L − gt2 (b) The chain will be completely on the ground when x(t) = L or t = 3 L/g . (c) The predicted velocity of the upper end of the chain when it hits the ground is infinity. 17. (a) The weight of x feet of the chain is 2x, so the corresponding mass is m = 2x/32 = x/16. The only force acting on the chain is the weight of the portion of the chain hanging over the edge of the platform. Thus, by Newton’s second law, ) ) ( ( ) ( d d x 1 dx dv 1 dv (mv) = v = +v + v 2 = 2x x = x dt dt 16 16 dt dt 16 dt and x dv/dt + v 2 = 32x. Now, by the Chain Rule, dv/dt = (dv/dx)(dx/dt) = v dv/dx, so xv dv/dx + v 2 = 32x. (b) We separate variables and write the differential equation as (v 2 − 32x) dx + xv dv = 0. This is not an exact form, but µ(x) = x is an integrating factor. Multiplying by x we get (xv 2 − 32x2 ) dx + x2 v dv = 0. This 1 2 2 32 3 3 form is the total differential of u = 12 x2 v 2 − 32 3 x , so an implicit solution is 2 x v − 3 x = c. Letting x = 3 and v = 0 we find c = −288. Solving for v we get √ dx 8 x3 − 27 √ =v= , 3 ≤ x ≤ 8. dt 3x (c) Separating variables and integrating we obtain √ x 8 dx = √ dt 3 x3 − 27 and ' 3 175 x √ s 8 ds = √ t + c. 3 s3 − 27 3.10 Nonlinear Models Since x = 3 when t = 0, we see that c = 0 and √ ' x 3 s √ t= ds. 3 8 3 s − 27 We want to find t when x = 7. Using a CAS we find t(7) = 0.576 seconds. 18. (a) There are two forces acting on the chain as it falls from the platform. One is the force due to gravity on the portion of the chain hanging over the edge of the platform. This is F1 = 2x. The second is due to the motion of the portion of the chain stretched out on the platform. By Newton’s second law this is + + * * d d (8 − x)2 d 8−x F2 = [mv] = v = v dt dt 32 dt 16 + * 8 − x dv 1 dx 1 dv = − v = − v2 . (8 − x) 16 dt 16 dt 16 dt From d [mv] = F1 − F2 we have dt + + * * 1 d 2x dv 2 v = 2x − −v (8 − x) dt 32 16 dt + * x dv 1 dx 1 dv 2 + v = 2x − −v (8 − x) 16 dt 16 dt 16 dt x dv dv + v 2 = 32x − (8 − x) + v2 dt dt x dv dv dv = 32x − 8 +x dt dt dt dv = 32x. dt By the Chain Rule, dv/dt = (dv/dx)(dx/dt) = v dv/dx, so 8 8 dv dv dv = 8v = 32x and v = 4x. dt dx dx (b) Integrating v dv = 4x dx we get 12 v 2 = 2x2 + c. Since v = 0 when x = 3, we have c = −18. Then √ v 2 = 4x2 − 36 and v = 4x2 − 36 . Using v = dx/dt, separating variables, and integrating we obtain dx x = 2 dt and cosh−1 = 2t + c1 . 2 3 x −9 Solving for x we get x(t) = 3 cosh(2t + c1 ). Since x = 3 when t = 0, we have cosh c1 = 1 and c1 = 0. Thus, x(t) = 3 cosh 2t. Differentiating, we find v(t) = dx/dt = 6 sinh 2t. √ (c) To find the time when the back end of the chain leaves the platform we solve x(t) = 3 cosh 2t = 8. This gives t1 = 1 2 cosh−1 8 3 ≈ 0.8184 seconds. The velocity at this instant is ( ) √ 8 v(t1 ) = 6 sinh cosh−1 = 2 55 ≈ 14.83 ft/s. 3 (d) Replacing 8 with L and 32 with g in part (a) we have L dv/dt = gx. Then dv dv dv g = Lv = gx and v = x. dt dx dx L Integrating we get 12 v 2 = (g/2L)x2 + c. Setting x = x0 and v = 0, we find c = −(g/2L)x20 . Solving for v L we find v(x) = 7 g 2 g 2 x − x0 . L L 176 3.10 Nonlinear Models Then the velocity at which the end of the chain leaves the edge of the platform is 7 g 2 v(L) = (L − x20 ) . L 19. Let (x, y) be the coordinates of S2 on the curve C. The slope at (x, y) is then dy/dx = (v1 t − y)/(0 − x) = (y − v1 t)/x or xy ′ − y = −v1 t. Differentiating with respect to x and using r = v1 /v2 gives dt dx dt ds ′′ xy = −v1 ds dx 3 1 xy ′′ = −v1 (− 1 + (y ′ )2 ) v2 3 ′′ xy = r 1 + (y ′ )2 . xy ′′ + y ′ − y ′ = −v1 Letting u = y ′ and separating variables, we obtain 3 du = r 1 + u2 dx du r √ = dx 2 x 1+u x sinh−1 u = r ln x + ln c = ln(cxr ) u = sinh(ln cxr ) ) ( dy 1 1 = cxr − r . dx 2 cx At t = 0, dy/dx = 0 and x = a, so 0 = car − 1/car . Thus c = 1/ar and * + dy 1 1# x $r # a $r 2 1 # x $r # x $−r − = − . = dx 2 a x 2 a a If r > 1 or r < 1, integrating gives y= * + 1 # x $1+r 1 # x $1−r a − + c1 . 2 1+r a 1−r a When t = 0, y = 0 and x = a, so 0 = (a/2)[1/(1 + r) − 1/(1 − r)] + c1 . Thus c1 = ar/(1 − r2 ) and * + a 1 # x $1+r 1 # x $1−r ar y= − . + 2 1+r a 1−r a 1 − r2 To see if the paths ever intersect we first note that if r > 1, then v1 > v2 and y → ∞ as x → 0+ . In other words, S2 always lags behind S1 . Next, if r < 1, then v1 < v2 and y = ar/(1 − r2 ) when x = 0. In other words, when the submarine’s speed is greater than the ship’s, their paths will intersect at the point (0, ar/(1 − r2 )). Finally, if r = 1, then integration gives + * 1 x2 1 y= − ln x + c2 . 2 2a a When t = 0, y = 0 and x = a, so 0 = (1/2)[a/2 − (1/a) ln a] + c2 . Thus c2 = −(1/2)[a/2 − (1/a) ln a] and + + * * * + 1 1 a 1 1 1 2 1 a 1 x2 − ln x − − ln a = (x − a2 ) + ln . y= 2 2a a 2 2 a 2 2a a x Since y → ∞ as x → 0+ , S2 will never catch up with S1 . 177 3.10 Nonlinear Models 20. (a) Let (r, θ) denote the polar coordinates of the destroyer S1 . When S1 travels the 6 miles from (9, 0) to (3, 0) it stands to reason, since S2 travels half as fast as S1 , that the polar coordinates of S2 are (3, θ2 ), where θ2 is unknown. In other words, the distances of the ships from (0, 0) are the same and r(t) = 15t then gives the radial distance of both ships. This is necessary if S1 is to intercept S2 . (b) The differential of arc length in polar coordinates is (ds)2 = (r dθ)2 + (dr)2 , so that ( ds dt )2 = r2 ( dθ dt )2 + ( dr dt )2 . Using ds/dt = 30 and dr/dt = 15 then gives 900 = 225t 2 675 = 225t2 ( ( dθ dt dθ dt )2 + 225 )2 √ dθ 3 = dt t √ √ r + c. θ(t) = 3 ln t + c = 3 ln 15 √ When r = 3, θ = 0, so c = − 3 ln 15 and θ(t) = Thus r = 3eθ/ √ 3 √ ) ( √ r r 1 − ln = 3 ln . 3 ln 15 5 3 , whose graph is a logarithmic spiral. (c) The time for S1 to go from (9, 0) to (3, 0) = 1 5 hour. Now S1 must intercept the path of S2 for some angle β, where 0 < β < 2π. At the time of interception t2 we have 15t2 = 3eβ/ is then t= √ 3 or t = 51 eβ/ √ 3 . The total time √ 1 1 β/√3 1 < (1 + e2π/ 3 ). + e 5 5 5 21. Since (dx/dt)2 is always positive, it is necessary to use |dx/dt|(dx/dt) in order to account for the fact that the motion is oscillatory and the velocity (or its square) should be negative when the spring is contracting. 22. (a) From the graph we see that the approximations appears to y be quite good for 0 ≤ x ≤ 0.4. Using an equation solver to solve sin x − x = 0.05 and sin x − x = 0.005, we find that the approximation is accurate to one decimal place for θ1 = 0.67 1.25 and to two decimal places for θ1 = 0.31. 0.75 1.5 1 0.5 0.25 0.25 0.5 0.75 178 1 1.25 1.5 x 3.10 Nonlinear Models (b) Θ Θ 1 1 Θ1 ! 1, 3, 5 0.5 1 2 3 Θ1 ! 7, 9, 11 0.5 4 5 6 t 1 -0.5 -0.5 -1 -1 23. (a) Write the differential equation as 2 3 4 5 6 θ moon 2 d2 θ + ω 2 sin θ = 0, dt2 earth 2 where ω = g/l. To test for differences between the earth t and the moon we take l = 3, θ(0) = 1, and θ′ (0) = 2. Using g = 32 on the earth and g = 5.5 on the moon we 5 -2 obtain the graphs shown in the figure. Comparing the apparent periods of the graphs, we see that the pendulum oscillates faster on the earth than on the moon. (b) The amplitude is greater on the moon than on the earth. (c) The linear model is θ d2 θ + ω 2 θ = 0, dt2 2 moon earth where ω 2 = g/l. When g = 32, l = 3, θ(0) = 1, and θ′ (0) = 2, the solution is t 5 θ(t) = cos 3.266t + 0.612 sin 3.266t. -2 When g = 5.5 the solution is θ(t) = cos 1.354t + 1.477 sin 1.354t. As in the nonlinear case, the pendulum oscillates faster on the earth than on the moon and still has greater amplitude on the moon. 24. (a) The general solution of d2 θ +θ =0 dt2 is θ(t) = c1 cos t + c2 sin t. From θ(0) = π/12 and θ′ (0) = −1/3 we find θ(t) = (π/12) cos t − (1/3) sin t. Setting θ(t) = 0 we have tan t = π/4 which implies t1 = tan−1 (π/4) ≈ 0.66577. (b) We set θ(t) = θ(0) + θ′ (0)t + 12 θ′′ (0)t2 + 16 θ′′′ (0)t3 + · · · and use θ′′ (t) = − sin θ(t) together with θ(0) = π/12 and θ′ (0) = −1/3. Then √ √ θ′′ (0) = − sin(π/12) = − 2 ( 3 − 1)/4 and √ √ θ′′′ (0) = − cos θ(0) · θ′ (0) = − cos(π/12)(−1/3) = 2 ( 3 + 1)/12. 179 3.10 Nonlinear Models Thus θ(t) = 1 π − t− 12 3 √ √ √ √ 2 ( 3 − 1) 2 2 ( 3 + 1) 3 t + t + ···. 8 72 (c) Setting π/12 − t/3 = 0 we obtain t1 = π/4 ≈ 0.785398. (d) Setting √ √ π 1 2 ( 3 − 1) 2 − t− t =0 12 3 8 and using the positive root we obtain t1 ≈ 0.63088. (e) Setting √ √ √ √ π 1 2 ( 3 − 1) 2 2 ( 3 + 1) 3 − t− t + t =0 12 3 8 72 we find with the help of a CAS that t1 ≈ 0.661973 is the first positive root. (f ) From the output we see that y(t) is an interpolating function on the 0.4 interval 0 ≤ t ≤ 5, whose graph is shown. The positive root of y(t) = 0 near t = 1 is t1 = 0.666404. 0.2 1 2 3 4 5 2 4 6 8 10 -0.2 -0.4 (g) To find the next two positive roots we change the interval used in NDSolve and Plot from {t,0,5} to {t,0,10}. We see from the graph that the second and third positive roots are near 4 and 7, respectively. Replacing {t,1} in FindRoot with {t,4} and then {t,7} we obtain t2 = 3.84411 and t3 = 7.0218. 0.4 0.2 -0.2 -0.4 25. From the table below we see that the pendulum first passes the vertical position between 1.7 and 1.8 seconds. To refine our estimate of t1 we estimate the solution of the differential equation on [1.7, 1.8] using a step size of h = 0.01. From the resulting table we see that t1 is between 1.76 and 1.77 seconds. Repeating the process with h = 0.001 we conclude that t1 ≈ 1.767. Then the period of the pendulum is approximately 4t1 = 7.068. The error when using t1 = 2π is 7.068 − 6.283 = 0.785 and the percentage relative error is (0.785/7.068)100 = 11.1. h=0.1 tn 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 h=0.01 θn 0.78540 0.78523 0.78407 0.78092 0.77482 0.76482 0.75004 0.72962 0.70275 0.66872 0.62687 0.57660 0.51744 0.44895 0.37085 0.28289 0.18497 0.07706 -0.04076 -0.16831 -0.30531 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 θn 0.07706 0.06572 0.05428 0.04275 0.03111 0.01938 0.00755 -0.00438 -0.01641 -0.02854 -0.04076 h=0.001 1.763 1.764 1.765 1.766 1.767 1.768 1.769 1.770 0.00398 0.00279 0.00160 0.00040 -0.00079 -0.00199 -0.00318 -0.00438 tn 180 3.11 Solving Systems of Linear Equations EXERCISES 3.11 Solving Systems of Linear Equations 1. From Dx = 2x − y and Dy = x we obtain y = 2x − Dx, Dy = 2Dx − D2 x, and (D2 − 2D + 1)x = 0. The solution is x = c1 et + c2 tet y = (c1 − c2 )et + c2 tet . 2. From Dx = 4x + 7y and Dy = x − 2y we obtain y = 17 Dx − 74 x, Dy = 17 D2 x − 74 Dx, and (D2 − 2D − 15)x = 0. The solution is x = c1 e5t + c2 e−3t 1 y = c1 e5t − c2 e−3t . 7 3. From Dx = −y + t and Dy = x − t we obtain y = t − Dx, Dy = 1 − D2 x, and (D2 + 1)x = 1 + t. The solution is x = c1 cos t + c2 sin t + 1 + t y = c1 sin t − c2 cos t + t − 1. 4. From Dx − 4y = 1 and x + Dy = 2 we obtain y = 41 Dx − 1 4 , Dy = 41 D2 x, and (D2 + 1)x = 2. The solution is x = c1 cos t + c2 sin t + 2 1 1 1 y = c2 cos t − c1 sin t − . 4 4 4 5. From (D2 + 5)x − 2y = 0 and −2x + (D2 + 2)y = 0 we obtain y = 2 1 2 2 (D + 5)x, D2 y = 1 4 2 (D + 5D2 )x, and 2 (D + 1)(D + 6)x = 0. The solution is √ √ x = c1 cos t + c2 sin t + c3 cos 6 t + c4 sin 6 t √ √ 1 1 y = 2c1 cos t + 2c2 sin t − c3 cos 6 t − c4 sin 6 t. 2 2 6. From (D + 1)x + (D − 1)y = 2 and 3x + (D + 2)y = −1 we obtain x = − 13 − 13 (D + 2)y, Dx = − 31 (D2 + 2D)y, and (D2 + 5)y = −7. The solution is √ √ 7 y = c1 cos 5 t + c2 sin 5 t − 5 / /√ 0 0 √ √ √ 2 3 2 5 5 x = − c1 − c2 cos 5 t + c1 − c2 sin 5 t + . 3 3 3 3 5 7. From D2 x = 4y + et and D2 y = 4x − et we obtain y = 41 D2 x − 41 et , D2 y = 14 D4 x − 41 et , and (D2 + 4)(D − 2)(D + 2)x = −3et . The solution is 1 x = c1 cos 2t + c2 sin 2t + c3 e2t + c4 e−2t + et 5 1 2t −2t y = −c1 cos 2t − c2 sin 2t + c3 e + c4 e − et . 5 181 3.11 Solving Systems of Linear Equations 8. From (D2 +5)x+Dy = 0 and (D +1)x+(D −4)y = 0 we obtain (D −5)(D2 +4)x = 0 and (D −5)(D2 +4)y = 0. The solution is x = c1 e5t + c2 cos 2t + c3 sin 2t y = c4 e5t + c5 cos 2t + c6 sin 2t. Substituting into (D + 1)x + (D − 4)y = 0 gives (6c1 + c4 )e5t + (c2 + 2c3 − 4c5 + 2c6 ) cos 2t + (−2c2 + c3 − 2c5 − 4c6 ) sin 2t = 0 so that c4 = −6c1 , c5 = 12 c3 , c6 = − 12 c2 , and 1 1 y = −6c1 e5t + c3 cos 2t − c2 sin 2t. 2 2 9. From Dx + D2 y = e3t and (D + 1)x + (D − 1)y = 4e3t we obtain D(D2 + 1)x = 34e3t and D(D2 + 1)y = −8e3t . The solution is 4 y = c1 + c2 sin t + c3 cos t − e3t 15 17 x = c4 + c5 sin t + c6 cos t + e3t . 15 Substituting into (D + 1)x + (D − 1)y = 4e3t gives (c4 − c1 ) + (c5 − c6 − c3 − c2 ) sin t + (c6 + c5 + c2 − c3 ) cos t = 0 so that c4 = c1 , c5 = c3 , c6 = −c2 , and x = c1 − c2 cos t + c3 sin t + 17 3t e . 15 10. From D2 x − Dy = t and (D + 3)x + (D + 3)y = 2 we obtain D(D + 1)(D + 3)x = 1 + 3t and D(D + 1)(D + 3)y = −1 − 3t. The solution is 1 x = c1 + c2 e−t + c3 e−3t − t + t2 2 1 y = c4 + c5 e−t + c6 e−3t + t − t2 . 2 Substituting into (D + 3)x + (D + 3)y = 2 and D2 x − Dy = t gives 3(c1 + c4 ) + 2(c2 + c5 )e−t = 2 and (c2 + c5 )e−t + 3(3c3 + c6 )e−3t = 0 so that c4 = −c1 , c5 = −c2 , c6 = −3c3 , and 1 y = −c1 − c2 e−t − 3c3 e−3t + t − t2 . 2 11. From (D2 − 1)x − y = 0 and (D − 1)x + Dy = 0 we obtain y = (D2 − 1)x, Dy = (D3 − D)x, and (D − 1)(D2 + D + 1)x = 0. The solution is 5 √ t −t/2 x = c1 e + e y= √ 6 3 3 c2 cos t + c3 sin t 2 2 0 0 /√ √ √ √ 3 3 3 3 3 3 −t/2 −t/2 − c2 − c3 e t+ c2 − c3 e t. cos sin 2 2 2 2 2 2 / 182 3.11 Solving Systems of Linear Equations 12. From (2D2 − D − 1)x − (2D + 1)y = 1 and (D − 1)x + Dy = −1 we obtain (2D + 1)(D − 1)(D + 1)x = −1 and (2D + 1)(D + 1)y = −2. The solution is x = c1 e−t/2 + c2 e−t + c3 et + 1 y = c4 e−t/2 + c5 e−t − 2. Substituting into (D − 1)x + Dy = −1 gives ) ( 1 3 − c1 − c4 e−t/2 + (−2c2 − c5 )e−t = 0 2 2 so that c4 = −3c1 , c5 = −2c2 , and y = −3c1 e−t/2 − 2c2 e−t − 2. 13. From (2D − 5)x + Dy = et and (D − 1)x + Dy = 5et we obtain Dy = (5 − 2D)x + et and (4 − D)x = 4et . Then 4 x = c1 e4t + et 3 and Dy = −3c1 e4t + 5et so that 3 y = − c1 e4t + c2 + 5et . 4 14. From Dx + Dy = et and (−D2 + D + 1)x + y = 0 we obtain y = (D2 − D − 1)x, Dy = (D3 − D2 − D)x, and D2 (D − 1)x = et . The solution is x = c1 + c2 t + c3 et + tet y = −c1 − c2 − c2 t − c3 et − tet + et . 15. Multiplying the first equation by D + 1 and the second equation by D2 + 1 and subtracting we obtain (D4 − D2 )x = 1. Then 1 x = c1 + c2 t + c3 et + c4 e−t − t2 . 2 Multiplying the first equation by D + 1 and subtracting we obtain D2 (D + 1)y = 1. Then 1 y = c5 + c6 t + c7 e−t − t2 . 2 Substituting into (D − 1)x + (D2 + 1)y = 1 gives (−c1 + c2 + c5 − 1) + (−2c4 + 2c7 )e−t + (−1 − c2 + c6 )t = 1 so that c5 = c1 − c2 + 2, c6 = c2 + 1, and c7 = c4 . The solution of the system is 1 x = c1 + c2 t + c3 et + c4 e−t − t2 2 1 y = (c1 − c2 + 2) + (c2 + 1)t + c4 e−t − t2 . 2 16. From D2 x−2(D2 +D)y = sin t and x+Dy = 0 we obtain x = −Dy, D2 x = −D3 y, and D(D2 +2D+2)y = − sin t. The solution is 1 2 y = c1 + c2 e−t cos t + c3 e−t sin t + cos t + sin t 5 5 1 2 x = (c2 + c3 )e−t sin t + (c2 − c3 )e−t cos t + sin t − cos t. 5 5 183 3.11 Solving Systems of Linear Equations 17. From Dx = y, Dy = z. and Dz = x we obtain x = D2 y = D3 x so that (D − 1)(D2 + D + 1)x = 0, 5 √ √ 6 3 3 x = c1 et + e−t/2 c2 sin t + c3 cos t , 2 2 0 0 /√ √ √ √ 3 3 3 3 1 1 −t/2 sin c3 e t+ c2 − c3 e−t/2 cos t, − c2 − 2 2 2 2 2 2 t / t / y = c1 e + and z = c1 e + 0 0 / √ √ √ √ 3 3 3 3 1 1 −t/2 − c2 + sin c3 e t+ − c2 − c3 e−t/2 cos t. 2 2 2 2 2 2 18. From Dx + z = et , (D − 1)x + Dy + Dz = 0, and x + 2y + Dz = et we obtain z = −Dx + et , Dz = −D2 x + et , and the system (−D2 + D − 1)x + Dy = −et and (−D2 + 1)x + 2y = 0. Then y = 12 (D2 − 1)x, Dy = 21 D(D2 − 1)x, and (D − 2)(D2 + 1)x = −2et so that the solution is x = c1 e2t + c2 cos t + c3 sin t + et 3 2t c1 e − c2 cos t − c3 sin t 2 z = −2c1 e2t − c3 cos t + c2 sin t. y= 19. Write the system in the form Dx − 6y = 0 x − Dy + z = 0 x + y − Dz = 0. Multiplying the second equation by D and adding to the third equation we obtain (D + 1)x − (D2 − 1)y = 0. Eliminating y between this equation and Dx − 6y = 0 we find (D3 − D − 6D − 6)x = (D + 1)(D + 2)(D − 3)x = 0. Thus x = c1 e−t + c2 e−2t + c3 e3t , and, successively substituting into the first and second equations, we get 1 1 1 y = − c1 e−t − c2 e−2t + c3 e3t 6 3 2 5 −t 1 −2t 1 3t + c3 e . z = − c1 e − c2 e 6 3 2 20. Write the system in the form (D + 1)x − z = 0 (D + 1)y − z = 0 x − y + Dz = 0. Multiplying the third equation by D + 1 and adding to the second equation we obtain (D +1)x+(D2 +D −1)z = 0. Eliminating z between this equation and (D +1)x−z = 0 we find D(D +1)2 x = 0. Thus x = c1 + c2 e−t + c3 te−t , and, successively substituting into the first and third equations, we get y = c1 + (c2 − c3 )e−t + c3 te−t z = c1 + c3 e−t . 184 3.11 Solving Systems of Linear Equations 21. From (D + 5)x + y = 0 and 4x − (D + 1)y = 0 we obtain y = −(D + 5)x so that Dy = −(D2 + 5D)x. Then 4x + (D2 + 5D)x + (D + 5)x = 0 and (D + 3)2 x = 0. Thus x = c1 e−3t + c2 te−3t y = −(2c1 + c2 )e−3t − 2c2 te−3t . Using x(1) = 0 and y(1) = 1 we obtain c1 e−3 + c2 e−3 = 0 −(2c1 + c2 )e−3 − 2c2 e−3 = 1 or c1 + c2 = 0 2c1 + 3c2 = −e3 . Thus c1 = e3 and c2 = −e3 . The solution of the initial value problem is x = e−3t+3 − te−3t+3 y = −e−3t+3 + 2te−3t+3 . 22. From Dx − y = −1 and 3x + (D − 2)y = 0 we obtain x = − 31 (D − 2)y so that Dx = − 13 (D2 − 2D)y. Then − 13 (D2 − 2D)y = y − 1 and (D2 − 2D + 3)y = 3. Thus # √ $ √ y = et c1 cos 2 t + c2 sin 2 t + 1 and #√ $ $ √ √ √ 2 2 1 t 1# e c1 − 2 c2 cos 2 t + 2 c1 + c2 sin 2 t + . 3 3 Using x(0) = y(0) = 0 we obtain c1 + 1 = 0 # $ 2 √ 1 c1 − 2 c2 + = 0. 3 3 √ Thus c1 = −1 and c2 = 2/2. The solution of the initial value problem is 0 / √ √ √ 2 2 2 t sin 2 t + x = e − cos 2 t − 3 6 3 / 0 √ √ √ 2 y = et − cos 2 t + sin 2 t + 1. 2 x= 23. Equating Newton’s law with the net forces in the x- and y-directions gives m d2 x/dt2 = 0 and m d2 y/dt2 = −mg, respectively. From mD2 x = 0 we obtain x(t) = c1 t + c2 , and from mD2 y = −mg or D2 y = −g we obtain y(t) = − 21 gt2 + c3 t + c4 . 24. From Newton’s second law in the x-direction we have d2 x 1 dx dx = −|c| . m 2 = −k cos θ = −k dt v dt dt In the y-direction we have d2 y 1 dy dy = −mg − k sin θ = −mg − k = −mg − |c| . 2 dt v dt dt From mD2 x + |c|Dx = 0 we have D(mD + |c|)x = 0 so that (mD + |c|)x = c or (D + |c|/m)x = c2 . This is a 1 & m linear first-order differential equation. An integrating factor is e |c|dt/m d |c|t/m [e x] = c2 e|c|t/m dt 185 = e|c|t/m so that 3.11 Solving Systems of Linear Equations and e|c|t/m x = (c2 m/|c|)e|c|t/m + c3 . The general solution of this equation is x(t) = c4 + c3 e−|c|t/m . From (mD2 +|c|D)y = −mg we have D(mD+|c|)y = −mg so that (mD+|c|)y = & −mgt+c1 or (D+|c|/m)y = −gt+c2 . This is a linear first-order differential equation with integrating factor e |c|dt/m = e|c|t/m . Thus d |c|t/m y] = (−gt + c2 )e|c|t/m [e dt mg |c|t/m m2 g |c|t/m e|c|t/m y = − te + 2 e + c3 e|c|t/m + c4 |c| c and y(t) = − mg m2 g t + 2 + c3 + c4 e−|c|t/m . |c| c 25. The FindRoot application of Mathematica gives a solution of x1 (t) = x2 (t) as approximately t = 13.73 minutes. So tank B contains more salt than tank A for t > 13.73 minutes. 26. (a) Separating variables in the first equation, we have dx1 /x1 = −dt/50, so x1 = c1 e−t/50 . From x1 (0) = 15 we get c1 = 15. The second differential equation then becomes dx2 15 −t/50 2 = e − x2 dt 50 75 dx2 2 3 −t/50 + x2 = e . dt 75 10 & This differential equation is linear and has the integrating factor e 2 dt/75 = e2t/75 . Then or 3 −t/50+2t/75 3 t/150 d 2t/75 [e x2 ] = e = e dt 10 10 so e2t/75 x2 = 45et/150 + c2 and x2 = 45e−t/50 + c2 e−2t/75 . From x2 (0) = 10 we get c2 = −35. The third differential equation then becomes or dx3 90 −t/50 70 −2t/75 1 = e − e − x3 dt 75 75 25 1 6 14 dx3 + x3 = e−t/50 − e−2t/75 . dt 25 5 15 & This differential equation is linear and has the integrating factor e dt/25 = et/25 . Then d t/25 6 14 6 14 x3 ] = e−t/50+t/25 − e−2t/75+t/25 = et/50 − et/75 , [e dt 5 15 5 15 so and et/25 x3 = 60et/50 − 70et/75 + c3 x3 = 60e−t/50 − 70e−2t/75 + c3 e−t/25 . From x3 (0) = 5 we get c3 = 15. The solution of the initial-value problem is x1 (t) = 15e−t/50 x2 (t) = 45e−t/50 − 35e−2t/75 x3 (t) = 60e−t/50 − 70e−2t/75 + 15e−t/25 . 186 3.11 Solving Systems of Linear Equations (b) pounds salt 14 12 10 8 6 4 2 x1 x2 x3 50 100 150 200 time (c) Solving x1 (t) = 21 , x2 (t) = 12 , and x3 (t) = 21 , FindRoot gives, respectively, t1 = 170.06 min, t2 = 214.7 min, and t3 = 224.4 min. Thus, all three tanks will contain less than or equal to 0.5 pounds of salt after 224.4 minutes. 27. (a) Write the system as (D2 + 3)x1 − x2 = 0 −2x1 + (D2 + 2)x2 = 0. Then (D2 + 2)(D2 + 3)x1 − 2x1 = (D2 + 1)(D2 + 4)x1 = 0, and x1 (t) = c1 cos t + c2 sin t + c3 cos 2t + c4 sin 2t. Since x2 = (D2 + 3)x1 , we have x2 (t) = 2c1 cos t + 2c2 sin t − c3 cos 2t − c4 sin 2t. The initial conditions imply x1 (0) = c1 + c3 = 2 x′1 (0) = c1 + 2c4 = 1 x2 (0) = 2c1 − c3 = −1 x′2 (0) = 2c2 − 2c4 = 1, so c1 = 1 3 , c2 = 2 3 , c3 = 5 3 , and c4 = 1 6 . Thus 1 cos t + 3 2 x2 (t) = cos t + 3 x1 (t) = (b) x1 3 2 1 -1 -2 -3 2 sin t + 3 4 sin t − 3 5 cos 2t + 3 5 cos 2t − 3 1 sin 2t 6 1 sin 2t. 6 x2 3 2 1 5 10 15 20 t -1 -2 -3 5 10 15 20 t In this problem the motion appears to be periodic with period 2π. In Figure 3.59 of Example 4 in the text the motion does not appear to be periodic. 187 3.11 Solving Systems of Linear Equations (c) x2 3 2 1 -2 -1 1 2 x1 -1 -2 CHAPTER 3 REVIEW EXERCISES 1. y = 0 2. Since yc = c1 ex + c2 e−x , a particular solution for y ′′ − y = 1 + ex is yp = A + Bxex . 3. It is not true unless the differential equation is homogeneous. For example, y1 = x is a solution of y ′′ + y = x, but y2 = 5x is not. 4. True 5. 8 ft, since k = 4 6. 2π/5, since 41 x′′ + 6.25x = 0 7. 5/4 m, since x = − cos 4t + 43 sin 4t √ √ 8. From x(0) = ( 2/2) sin φ = −1/2 we see that sin φ = −1/ 2 , so φ is an angle in the third or fourth quadrant. √ √ Since x′ (t) = 2 cos(2t + φ), x′ (0) = 2 cos φ = 1 and cos φ > 0. Thus φ is in the fourth quadrant and φ = −π/4. 9. The set is linearly independent over (−∞, ∞) and linearly dependent over (0, ∞). 10. (a) Since f2 (x) = 2 ln x = 2f1 (x), the set of functions is linearly dependent. (b) Since xn+1 is not a constant multiple of xn , the set of functions is linearly independent. (c) Since x + 1 is not a constant multiple of x, the set of functions is linearly independent. (d) Since f1 (x) = cos x cos(π/2) − sin x sin(π/2) = − sin x = −f2 (x), the set of functions is linearly dependent. (e) Since f1 (x) = 0 · f2 (x), the set of functions is linearly dependent. (f ) Since 2x is not a constant multiple of 2, the set of functions is linearly independent. (g) Since 3(x2 ) + 2(1 − x2 ) − (2 + x2 ) = 0, the set of functions is linearly dependent. (h) Since xex+1 + 0(4x − 5)ex − exex = 0, the set of functions is linearly dependent. 188 CHAPTER 3 REVIEW EXERCISES 11. (a) The auxiliary equation is (m − 3)(m + 5)(m − 1) = m3 + m2 − 17m + 15 = 0, so the differential equation is y ′′′ + y ′′ − 17y ′ + 15y = 0. (b) The form of the auxiliary equation is m(m − 1)(m − 2) + bm(m − 1) + cm + d = m3 + (b − 3)m2 + (c − b + 2)m + d = 0. Since (m − 3)(m + 5)(m − 1) = m3 + m2 − 17m + 15 = 0, we have b − 3 = 1, c − b + 2 = −17, and d = 15. Thus, b = 4 and c = −15, so the differential equation is y ′′′ + 4y ′′ − 15y ′ + 15y = 0. 12. (a) The auxiliary equation is am(m − 1) + bm + c = am2 + (b − a)m + c = 0. If the roots are 3 and −1, then we want (m − 3)(m + 1) = m2 − 2m − 3 = 0. Thus, let a = 1, b = −1, and c = −3, so that the differential equation is x2 y ′′ − xy ′ − 3y = 0. (b) In this case we want the auxiliary equation to be m2 + 1 = 0, so let a = 1, b = 1, and c = 1. Then the differential equation is x2 y ′′ + xy ′ + y = 0. √ 13. From m2 − 2m − 2 = 0 we obtain m = 1 ± 3 so that y = c1 e(1+ √ 3 )x √ + c2 e(1− 3 )x . √ 14. From 2m2 + 2m + 3 = 0 we obtain m = −1/2 ± ( 5/2)i so that / √ √ 0 5 5 −x/2 c1 cos y=e x + c2 sin x . 2 2 15. From m3 + 10m2 + 25m = 0 we obtain m = 0, m = −5, and m = −5 so that y = c1 + c2 e−5x + c3 xe−5x . 16. From 2m3 + 9m2 + 12m + 5 = 0 we obtain m = −1, m = −1, and m = −5/2 so that y = c1 e−5x/2 + c2 e−x + c3 xe−x . √ 17. From 3m3 + 10m2 + 15m + 4 = 0 we obtain m = −1/3 and m = −3/2 ± ( 7/2)i so that / √ 0 √ 7 7 −x/3 −3x/2 y = c1 e x + c3 sin x . +e c2 cos 2 2 √ 18. From 2m4 + 3m3 + 2m2 + 6m − 4 = 0 we obtain m = 1/2, m = −2, and m = ± 2 i so that √ √ y = c1 ex/2 + c2 e−2x + c3 cos 2 x + c4 sin 2 x. 19. Applying D4 to the differential equation we obtain D4 (D2 − 3D + 5) = 0. Then 0 / √ √ 11 11 3x/2 y=e c1 cos x + c2 sin x + c3 + c4 x + c5 x2 + c6 x3 2 2 :; < 9 yc 2 3 and yp = A + Bx + Cx + Dx . Substituting yp into the differential equation yields (5A − 3B + 2C) + (5B − 6C + 6D)x + (5C − 9D)x2 + 5Dx3 = −2x + 4x3 . Equating coefficients gives A = −222/625, B = 46/125, C = 36/25, and D = 4/5. The general solution is / 0 √ √ 11 11 222 46 36 4 y = e3x/2 c1 cos x + c2 sin x − + x + x2 + x3 . 2 2 625 125 25 5 189 CHAPTER 3 REVIEW EXERCISES 20. Applying (D − 1)3 to the differential equation we obtain (D − 1)3 (D − 2D + 1) = (D − 1)5 = 0. Then y = c1 ex + c2 xex + c3 x2 ex + c4 x3 ex + c5 x4 ex 9 :; < yc 2 x 3 x 4 x and yp = Ax e + Bx e + Cx e . Substituting yp into the differential equation yields 12Cx2 ex + 6Bxex + 2Aex = x2 ex . Equating coefficients gives A = 0, B = 0, and C = 1/12. The general solution is y = c1 ex + c2 xex + 1 4 x x e . 12 21. Applying D(D2 + 1) to the differential equation we obtain D(D2 + 1)(D3 − 5D2 + 6D) = D2 (D2 + 1)(D − 2)(D − 3) = 0. Then y = c1 + c2 e2x + c3 e3x + c4 x + c5 cos x + c6 sin x :; < 9 yc and yp = Ax + B cos x + C sin x. Substituting yp into the differential equation yields 6A + (5B + 5C) cos x + (−5B + 5C) sin x = 8 + 2 sin x. Equating coefficients gives A = 4/3, B = −1/5, and C = 1/5. The general solution is 4 1 1 y = c1 + c2 e2x + c3 e3x + x − cos x + sin x. 3 5 5 22. Applying D to the differential equation we obtain D(D3 − D2 ) = D3 (D − 1) = 0. Then y = c1 + c2 x + c3 ex + c4 x2 9 :; < yc and yp = Ax2 . Substituting yp into the differential equation yields −2A = 6. Equating coefficients gives A = −3. The general solution is y = c1 + c2 x + c3 ex − 3x2 . 23. The auxiliary equation is m2 − 2m + 2 = [m − (1 + i)][m − (1 − i)] = 0, so yc = c1 ex sin x + c2 ex cos x and % % % % ex sin x ex cos x % % = −e2x . W =% x e cos x + ex sin x −ex sin x + ex cos x % Identifying f (x) = ex tan x we obtain u′1 = − u′2 = (ex cos x)(ex tan x) = sin x −e2x sin2 x (ex sin x)(ex tan x) = − = cos x − sec x. −e2x cos x Then u1 = − cos x, u2 = sin x − ln | sec x + tan x|, and y = c1 ex sin x + c2 ex cos x − ex sin x cos x + ex sin x cos x − ex cos x ln | sec x + tan x| = c1 ex sin x + c2 ex cos x − ex cos x ln | sec x + tan x|. 24. The auxiliary equation is m2 − 1 = 0, so yc = c1 ex + c2 e−x and % x % %e e−x % % = −2. W = %% x e −e−x % 190 CHAPTER 3 REVIEW EXERCISES Identifying f (x) = 2ex /(ex + e−x ) we obtain u′1 = 1 ex = ex + e−x 1 + e2x u′2 = − e2x e3x ex x = − = −e + . ex + e−x 1 + e2x 1 + e2x Then u1 = tan−1 ex , u2 = −ex + tan−1 ex , and y = c1 ex + c2 e−x + ex tan−1 ex − 1 + e−x tan−1 ex . 25. The auxiliary equation is 6m2 − m − 1 = 0 so that y = c1 x1/2 + c2 x−1/3 . 26. The auxiliary equation is 2m3 + 13m2 + 24m + 9 = (m + 3)2 (m + 1/2) = 0 so that y = c1 x−3 + c2 x−3 ln x + c3 x−1/2 . 27. The auxiliary equation is m2 − 5m + 6 = (m − 2)(m − 3) = 0 and a particular solution is yp = x4 − x2 ln x so that y = c1 x2 + c2 x3 + x4 − x2 ln x. 28. The auxiliary equation is m2 − 2m + 1 = (m − 1)2 = 0 and a particular solution is yp = 41 x3 so that 1 y = c1 x + c2 x ln x + x3 . 4 29. (a) The auxiliary equation is m2 + ω 2 = 0, so yc = c1 cos ωt + c2 sin ωt. When ω ̸= α, yp = A cos αt + B sin αt and y = c1 cos ωt + c2 sin ωt + A cos αt + B sin αt. When ω = α, yp = At cos ωt + Bt sin ωt and y = c1 cos ωt + c2 sin ωt + At cos ωt + Bt sin ωt. (b) The auxiliary equation is m2 − ω 2 = 0, so yc = c1 eωt + c2 e−ωt . When ω ̸= α, yp = Aeαt and y = c1 eωt + c2 e−ωt + Aeαt . When ω = α, yp = Ateωt and y = c1 eωt + c2 e−ωt + Ateωt . 30. (a) If y = sin x is a solution then so is y = cos x and m2 + 1 is a factor of the auxiliary equation m4 + 2m3 + 11m2 + 2m + 10 = 0. Dividing by m2 + 1 we get m2 + 2m + 10, which has roots −1 ± 3i. The general solution of the differential equation is y = c1 cos x + c2 sin x + e−x (c3 cos 3x + c4 sin 3x). (b) The auxiliary equation is m(m + 1) = m2 + m = 0, so the associated homogeneous differential equation is y ′′ + y ′ = 0. Letting y = c1 + c2 e−x + 21 x2 − x and computing y ′′ + y ′ we get x. Thus, the differential equation is y ′′ + y ′ = x. 31. (a) The auxiliary equation is m4 − 2m2 + 1 = (m2 − 1)2 = 0, so the general solution of the differential equation is y = c1 sinh x + c2 cosh x + c3 x sinh x + c4 x cosh x. 191 CHAPTER 3 REVIEW EXERCISES (b) Since both sinh x and x sinh x are solutions of the associated homogeneous differential equation, a particular solution of y (4) − 2y ′′ + y = sinh x has the form yp = Ax2 sinh x + Bx2 cosh x. 32. Since y1′ = 1 and y1′′ = 0, x2 y1′′ − (x2 + 2x)y1′ + (x + 2)y1 = −x2 − 2x + x2 + 2x = 0, and y1 = x is a solution of the associated homogeneous equation. Using the method of reduction of order, we let y = ux. Then y ′ = xu′ + u and y ′′ = xu′′ + 2u′ , so x2 y ′′ − (x2 + 2x)y ′ + (x + 2)y = x3 u′′ + 2x2 u′ − x3 u′ − 2x2 u′ − x2 u − 2xu + x2 u + 2xu = x3 u′′ − x3 u′ = x3 (u′′ − u′ ). To find a second solution of the homogeneous equation we note that u = ex is a solution of u′′ − u′ = 0. Thus, yc = c1 x + c2 xex . To find a particular solution we set x3 (u′′ − u′ ) = x3 so that u′′ − u′ = 1. This differential equation has a particular solution of the form Ax. Substituting, we find A = −1, so a particular solution of the original differential equation is yp = −x2 and the general solution is y = c1 x + c2 xex − x2 . 33. The auxiliary equation is m2 − 2m + 2 = 0 so that m = 1 ± i and y = ex (c1 cos x + c2 sin x). Setting y(π/2) = 0 and y(π) = −1 we obtain c1 = e−π and c2 = 0. Thus, y = ex−π cos x. 34. The auxiliary equation is m2 + 2m + 1 = (m + 1)2 = 0, so that y = c1 e−x + c2 xe−x . Setting y(−1) = 0 and y ′ (0) = 0 we get c1 e − c2 e = 0 and −c1 + c2 = 0. Thus c1 = c2 and y = c1 (e−x + xe−x ) is a solution of the boundary-value problem for any real number c1 . 35. The auxiliary equation is m2 − 1 = (m − 1)(m + 1) = 0 so that m = ±1 and y = c1 ex + c2 e−x . Assuming yp = Ax + B + C sin x and substituting into the differential equation we find A = −1, B = 0, and C = − 21 . Thus yp = −x − 1 2 sin x and y = c1 ex + c2 e−x − x − 1 sin x. 2 Setting y(0) = 2 and y ′ (0) = 3 we obtain Solving this system we find c1 = 13 4 c1 + c2 = 2 3 c1 − c2 − = 3. 2 5 and c2 = − 4 . The solution of the initial-value problem is y= 1 13 x 5 −x e − e − x − sin x. 4 4 2 36. The auxiliary equation is m2 + 1 = 0, so yc = c1 cos x + c2 sin x and % % % cos x sin x % % % = 1. W =% − sin x cos x % Identifying f (x) = sec3 x we obtain u′1 = − sin x sec3 x = − sin x cos3 x u′2 = cos x sec3 x = sec2 x. Then u1 = − 1 1 1 = − sec2 x 2 cos2 x 2 u2 = tan x. Thus y = c1 cos x + c2 sin x − 1 cos x sec2 x + sin x tan x 2 192 CHAPTER 3 REVIEW EXERCISES = c1 cos x + c2 sin x − 1 1 − cos2 x sec x + 2 cos x = c3 cos x + c2 sin x + 1 sec x. 2 and y ′ = −c3 sin x + c2 cos x + 1 sec x tan x. 2 The initial conditions imply c3 + 1 =1 2 1 c2 = . 2 Thus c3 = c2 = 1/2 and y= 1 1 1 cos x + sin x + sec x. 2 2 2 37. Let u = y ′ so that u′ = y ′′ . The equation becomes u du/dx = 4x. Separating variables we obtain 1 u du = 4x dx =⇒ u2 = 2x2 + c1 =⇒ u2 = 4x2 + c2 . 2 When x = 1, y ′ = u = 2, so 4 = 4 + c2 and c2 = 0. Then dy dy = 2x or = −2x u2 = 4x2 =⇒ dx dx =⇒ y = x2 + c3 or y = −x2 + c4 . When x = 1, y = 5, so 5 = 1 + c3 and 5 = −1 + c4 . Thus c3 = 4 and c4 = 6. We have y = x2 + 4 and y = −x2 + 6. Note however that when y = −x2 + 6, y ′ = −2x and y ′ (1) = −2 ̸= 2. Thus, the solution of the initial-value problem is y = x2 + 4. 38. Let u = y ′ so that y ′′ = u du/dy. The equation becomes 2u du/dy = 3y 2 . Separating variables we obtain 2u du = 3y 2 dy =⇒ u2 = y 3 + c1 . When x = 0, y = 1 and y ′ = u = 1 so 1 = 1 + c1 and c1 = 0. Then ( )2 dy dy 2 3 u = y =⇒ = y 3 =⇒ = y 3/2 =⇒ y −3/2 dy = dx dx dx 4 =⇒ −2y −1/2 = x + c2 =⇒ y = . (x + c2 )2 When x = 0, y = 1, so 1 = 4/c22 and c2 = ±2. Thus, y = 4/(x + 2)2 and y = 4/(x − 2)2 . Note, however, that when y = 4/(x + 2)2 , y ′ = −8/(x + 2)3 and y ′ (0) = −1 ̸= 1. Thus, the solution of the initial-value problem is y = 4/(x − 2)2 . 39. (a) The auxiliary equation is 12m4 + 64m3 + 59m2 − 23m − 12 = 0 and has roots −4, − 32 , − 31 , and general solution is y = c1 e−4x + c2 e−3x/2 + c3 e−x/3 + c4 ex/2 . (b) The system of equations is c1 + c2 + c3 + c4 3 1 1 −4c1 − c2 − c3 + c4 2 3 2 1 1 9 16c1 + c2 + c3 + c4 4 9 4 27 1 1 −64c1 − c2 − c3 + c4 8 27 8 193 = −1 =2 =5 = 0. 1 2. The CHAPTER 3 REVIEW EXERCISES 73 Using a CAS we find c1 = − 495 , c2 = 109 35 problem is y=− , c3 = − 3726 385 , and c4 = 257 45 . The solution of the initial-value 73 −4x 109 −3x/2 3726 −x/3 257 x/2 e + e − e + e . 495 35 385 45 40. Consider xy ′′ + y ′ = 0 and look for a solution of the form y = xm . Substituting y into the differential equation we have 1 xy ′′ + y ′ = m(m − 1)xm−1 + mxm−1 = m2 xm−1 . Identifying f (x) = −x 3 4 5 x -1 Thus, the general solution of xy ′′ +y ′ = 0 is yc = c1 +c2 ln x. To find a particular √ solution of xy ′′ + y ′ = − x we use variation of parameters. The Wronskian is % % % 1 ln x % 1 %= . % W =% 0 1/x % x −1/2 2 -2 -3 -4 -5 we obtain u′1 = √ x−1/2 ln x √ −x−1/2 = x ln x and u′2 = = − x, 1/x 1/x so that u1 = x3/2 Then #2 3 ln x − 4$ 9 2 and u2 = − x3/2 . 3 !2 4 " 2 3/2 4 ln x − − x ln x = − x3/2 3 9 3 9 and the general solution of the differential equation is yp = x3/2 4 y = c1 + c2 ln x − x3/2 . 9 The initial conditions are y(1) = 0 and y ′ (1) = 0. These imply that c1 = 4 9 and c2 = 2 3 . The solution of the initial-value problem is y= 4 2 4 + ln x − x3/2 . 9 3 9 The graph is shown above. 41. From (D − 2)x + (D − 2)y = 1 and Dx + (2D − 1)y = 3 we obtain (D − 1)(D − 2)y = −6 and Dx = 3 − (2D − 1)y. Then 3 y = c1 e2t + c2 et − 3 and x = −c2 et − c1 e2t + c3 . 2 Substituting into (D − 2)x + (D − 2)y = 1 gives c3 = 52 so that 5 3 x = −c2 et − c1 e2t + . 2 2 42. From (D − 2)x − y = t − 2 and −3x + (D − 4)y = −4t we obtain (D − 1)(D − 5)x = 9 − 8t. Then and 8 3 x = c1 et + c2 e5t − t − 5 25 y = (D − 2)x − t + 2 = −c1 et + 3c2 e5t + 16 11 + t. 25 25 43. From (D − 2)x − y = −et and −3x + (D − 4)y = −7et we obtain (D − 1)(D − 5)x = −4et so that x = c1 et + c2 e5t + tet . Then 194 CHAPTER 3 REVIEW EXERCISES y = (D − 2)x + et = −c1 et + 3c2 e5t − tet + 2et . 44. From (D + 2)x + (D + 1)y = sin 2t and 5x + (D + 3)y = cos 2t we obtain (D2 + 5)y = 2 cos 2t − 7 sin 2t. Then and y = c1 cos t + c2 sin t − 2 7 cos 2t + sin 2t 3 3 1 1 x = − (D + 3)y + cos 2t 5 5 ( ) ( ) 1 1 5 3 3 1 = c1 − c2 sin t + − c2 − c1 cos t − sin 2t − cos 2t. 5 5 5 5 3 3 45. The period of a spring/mass system is given by T = 2π/ω where ω 2 = k/m = kg/W , where k is the spring constant, W is the weight of the mass attached to the spring, and g is the acceleration due to gravity. Thus, √ √ √ √ the period of oscillation is T = (2π/ kg ) W . If the weight of the original mass is W , then (2π/ kg ) W = 3 √ √ √ √ and (2π/ kg ) W − 8 = 2. Dividing, we get W / W − 8 = 3/2 or W = 49 (W − 8). Solving for W we find that the weight of the original mass was 14.4 pounds. 46. (a) Solving 38 x′′ + 6x = 0 subject to x(0) = 1 and x′ (0) = −4 we obtain √ x = cos 4t − sin 4t = 2 sin (4t + 3π/4) . (b) The amplitude is √ 2, period is π/2, and frequency is 2/π. (c) If x = 1 then t = nπ/2 and t = −π/8 + nπ/2 for n = 1, 2, 3, . . . . (d) If x = 0 then t = π/16 + nπ/4 for n = 0, 1, 2, . . .. The motion is upward for n even and downward for n odd. (e) x′ (3π/16) = 0 (f ) If x′ = 0 then 4t + 3π/4 = π/2 + nπ or t = 3π/16 + nπ. 47. From mx′′ + 4x′ + 2x = 0 we see that nonoscillatory motion results if 16 − 8m ≥ 0 or 0 < m ≤ 2. 48. From x′′ + βx′ + 64x = 0 we see that oscillatory motion results if β 2 − 256 < 0 or 0 ≤ β < 16. 49. From q ′′ + 104 q = 100 sin 50t, q(0) = 0, and q ′ (0) = 0 we obtain qc = c1 cos 100t + c2 sin 100t, qp = and 1 sin 100t + (a) q = − 150 (b) i = − 32 cos 100t + 2 3 1 75 sin 50t, cos 50t, and (c) q = 0 when sin 50t(1 − cos 50t) = 0 or t = nπ/50 for n = 0, 1, 2, . . . . 50. By Kirchhoff’s second law, d2 q dq 1 +R + q = E(t). dt2 dt C Using q ′ (t) = i(t) we can write the differential equation in the form L L 1 di + Ri + q = E(t). dt C Then differentiating we obtain L d2 i di 1 +R + i = E ′ (t). dt2 dt C 51. For λ = α2 > 0 the general solution is y = c1 cos αx + c2 sin αx. Now y(0) = c1 and y(2π) = c1 cos 2πα + c2 sin 2πα, 195 1 75 sin 50t, CHAPTER 3 REVIEW EXERCISES so the condition y(0) = y(2π) implies c1 = c1 cos 2πα + c2 sin 2πα which is true when α = √ λ = n or λ = n2 for n = 1, 2, 3, . . . . Since y ′ = −αc1 sin αx + αc2 cos αx = −nc1 sin nx + nc2 cos nx, we see that y ′ (0) = nc2 = y ′ (2π) for n = 1, 2, 3, . . . . Thus, the eigenvalues are n2 for n = 1, 2, 3, . . . , with corresponding eigenfunctions cos nx and sin nx. When λ = 0, the general solution is y = c1 x + c2 and the corresponding eigenfunction is y = 1. For λ = −α2 < 0 the general solution is y = c1 cosh αx + c2 sinh αx. In this case y(0) = c1 and y(2π) = c1 cosh 2πα + c2 sinh 2πα, so y(0) = y(2π) can only be valid for α = 0. Thus, there are no eigenvalues corresponding to λ < 0. 52. (a) The differential equation is d2 r/dt2 − ω 2 r = −g sin ωt. The auxiliary equation is m2 − ω 2 = 0, so rc = c1 eωt + c2 e−ωt . A particular solution has the form rp = A sin ωt + B cos ωt. Substituting into the differential equation we find −2Aω 2 sin ωt − 2Bω 2 cos ωt = −g sin ωt. Thus, B = 0, A = g/2ω 2 , and rp = (g/2ω 2 ) sin ωt. The general solution of the differential equation is r(t) = c1 eωt +c2 e−ωt +(g/2ω 2 ) sin ωt. The initial conditions imply c1 + c2 = r0 and g/2ω − ωc1 + ωc2 = v0 . Solving for c1 and c2 we get c1 = (2ω 2 r0 + 2ωv0 − g)/4ω 2 and c2 = (2ω 2 r0 − 2ωv0 + g)/4ω 2 , so that r(t) = 2ω 2 r0 + 2ωv0 − g ωt 2ω 2 r0 − 2ωv0 + g −ωt g e + e + sin ωt. 4ω 2 4ω 2 2ω 2 (b) The bead will exhibit simple harmonic motion when the exponential terms are missing. Solving c1 = 0, c2 = 0 for r0 and v0 we find r0 = 0 and v0 = g/2ω. To find the minimum length of rod that will accommodate simple harmonic motion we determine the amplitude of r(t) and double it. Thus L = g/ω 2 . (c) As t increases, eωt approaches infinity and e−ωt approaches 0. Since sin ωt is bounded, the distance, r(t), of the bead from the pivot point increases without bound and the distance of the bead from P will eventually exceed L/2. (d) r 17 20 16.1 16 10 2 4 6 8 10 12 14 -10 -20 0 10 15 196 t CHAPTER 3 REVIEW EXERCISES (e) For each v0 we want to find the smallest value of t for which r(t) = ±20. Whether we look for r(t) = −20 or r(t) = 20 is determined by looking at the graphs in part (d). The total times that the bead stays on the rod is shown in the table below. v0 0 10 15 16.1 17 r t -20 1.55007 -20 2.35494 -20 3.43088 20 6.11627 20 4.22339 When v0 = 16 the bead never leaves the rod. 53. Unlike the derivation given in Section 3.8 in the text, the weight mg of the mass m does not appear in the net force since the spring is not stretched by the weight of the mass when it is in the equilibrium position (i.e. there is no mg − ks term in the net force). The only force acting on the mass when it is in motion is the restoring force of the spring. By Newton’s second law, m d2 x = −kx dt2 or d2 x k + x = 0. dt2 m 54. The force of kinetic friction opposing the motion of the mass in µN , where µ is the coefficient of sliding friction and N is the normal component of the weight. Since friction is a force opposite to the direction of motion and since N is pointed directly downward (it is simply the weight of the mass), Newton’s second law gives, for motion to the right (x′ > 0) , m d2 x = −kx − µmg, dt2 and for motion to the left (x′ < 0), d2 x = −kx + µmg. dt2 Traditionally, these two equations are written as one expression m m d2 x + fx sgn(x′ ) + kx = 0, dt2 where fk = µmg and ′ sgn(x ) = = 1, −1, 197 x′ > 0 x′ < 0. 4 The Laplace Transform EXERCISES 4.1 Definition of the Laplace Transform 1. 2. 3. 4. 5. 6. 7. 8. "1 "∞ 1 −st "" 1 −st "" e dt = e −e dt + {f (t)} = " −se " s 1 0 0 1 $ # 2 −s 1 1 −s 1 −s 1 = e − , s>0 = e − − 0− e s s s s s "2 ! 2 " 4 4 4e−st dt = − e−st "" = − (e−2s − 1), s > 0 {f (t)} = s s 0 0 "∞ # $ "1 ! ∞ ! 1 1 −st "" 1 −st 1 −st "" −st −st e dt = − te te dt + {f (t)} = − 2e " −se " s s 1 0 0 1 # $ # $ 1 −s 1 −s 1 1 1 −s −s = − e − 2e − 0 − 2 − (0 − e ) = 2 (1 − e ), s > 0 s s s s s # $ "1 ! 1 " 2 2 1 {f (t)} = (2t + 1)e−st dt = − te−st − 2 e−st − e−st "" s s s 0 0 # $ # $ 2 2 2 1 1 2 1 = − e−s − 2 e−s − e−s − 0 − 2 − = (1 − 3e−s ) + 2 (1 − e−s ), s s s s s s s # $ "π ! π " s 1 {f (t)} = (sin t)e−st dt = − 2 e−st sin t − 2 e−st cos t "" s +1 s +1 0 0 $ $ # # 1 1 1 e−πs − 0 − 2 = 2 (e−πs + 1), s > 0 = 0+ 2 s +1 s +1 s +1 $ "∞ # ! ∞ " 1 s −st −st −st e cos t + 2 e sin t "" (cos t)e dt = − 2 {f (t)} = s + 1 s + 1 π/2 π/2 # $ 1 1 =0− 0+ 2 e−πs/2 = − 2 e−πs/2 , s > 0 s +1 s +1 % 0, 0 < t < 1 f (t) = t, t > 1 # $ "∞ ! ∞ 1 −s 1 −st 1 −st "" 1 −s −st te dt = − te {f (t)} = − 2e " = s e + s2 e , s > 0 s s 1 1 % 0, 0<t<1 f (t) = 2t − 2, t > 1 $ "∞ # ! ∞ " 1 1 2 {f (t)} = 2 (t − 1)e−st dt = 2 − (t − 1)e−st − 2 e−st "" = 2 e−s , s > 0 s s s ! 1 −st ! ∞ −st 1 1 198 s>0 4.1 Definition of the Laplace Transform 0<t<1 so t>1 # $ "1 ! 1 ! ∞ ! 1 1 1 −st "" −st −st −st −st {f (t)} = (1 − t)e dt = − (1 − t)e 0e dt = (1 − t)e dt + + 2e " s s 0 1 0 0 1 1 −s 1 = 2e + − 2, s>0 s s s 9. The function is f (t) = 1 − t, 0, % ⎧ ⎨ 0, 0 < t < a 10. f (t) = c, a < t < b ; ⎩ 0, t > b 11. {f (t)} = ! ∞ ! ∞ ! ∞ {f (t)} = e 7 dt = e {f (t)} = ∞ ! −2t−5 −st e e −5 dt = e 0 13. {f (t)} = 4t −st te e ∞ ! = # dt = {f (t)} = ! ! ∞ ! "∞ e7 (1−s)t "" e7 e7 dt = = 0 − e = , " 1−s 1−s s−1 0 −(s+2)t e ∞ (4−s)t te "b c −sa c −st "" − e−sb ), s > 0 dt = − e " = s (e s a dt = # "∞ e−5 −(s+2)t "" e−5 dt = − e , = " s+2 s+2 0 1 1 te(4−s)t − e(4−s)t 4−s (4 − s)2 s>4 t2 e−2t e−st dt = ! ∞ − ∞ e−t (sin t)e−st dt = ! ∞ {f (t)} = # ∞ ! et (cos t)e−st dt = # (sin t)e−(s+1)t dt ! ∞ 17. {f (t)} = $ "∞ " 2 " = , " (s + 2)3 0 $ "∞ " " " 0 (cos t)e(1−s)t dt 1−s 1 e(1−s)t cos t + e(1−s)t sin t (1 − s)2 + 1 (1 − s)2 + 1 s−1 1−s =− = 2 , s>1 (1 − s)2 + 1 s − 2s + 2 ! 0 0 0 = $ "∞ " " " 0 −(s + 1) −(s+1)t 1 e sin t − e−(s+1)t cos t (s + 1)2 + 1 (s + 1)2 + 1 1 1 = 2 , s > −1 = (s + 1)2 + 1 s + 2s + 2 16. s > −2 t2 e−(s+2)t dt 2 2 1 2 −(s+2)t te−(s+2)t − e−(s+2)t t e − s+2 (s + 2)2 (s + 2)3 0 = ∞ $ "∞ " " " 0 t(cos t)e−st dt 0 = = s>1 0 0 15. e 0 1 = , (4 − s)2 {f (t)} = (1−s)t 0 0 14. ce 0 0 12. −st a t+7 −st e b ! )* s2 − 1 st − 2 − s + 1 (s2 + 1)2 s2 − 1 2 (s2 + 1) , + (cos t)e−st + * 2s t + s2 + 1 (s2 + 1)2 s>0 199 + (sin t)e−st ,∞ 0 s > −2 4.1 18. Definition of the Laplace Transform {f (t)} = ! ∞ t(sin t)e−st dt 0 = = )* 2s t − − 2 2 s + 1 (s + 1)2 2s , 2 (s2 + 1) (sin t)e ,∞ 0 s>0 5! s6 24. {−4t2 + 16t + 9} = −4 26. {8t3 − 12t2 + 6t − 1} = 8 28. {t2 − e−9t + 5} = 2 1 1 + + s s−2 s−4 30. {e2t − 2 + e−2t } = 1 2 1 − + s−2 s s+2 2 3 −5 2 s3 s +9 32. {cos 5t + sin 2t} = s 2 + s2 + 25 s2 + 4 {t2 + 6t − 3} = 25. {t3 + 3t2 + 3t + 1} = 27. {1 + e4t } = 29. {1 + 2e2t + e4t } = 31. {4t2 − 5 sin 3t} = 4 33. 1 {sinh kt} = 2 34. {cosh kt} = 38. −st {7t + 3} = 23. 37. + 22. {4t − 10} = 36. st s2 − 1 + 2 s + 1 (s2 + 1)2 {t5 } = 21. 4 10 − s2 s 2 6 3 + 2− s3 s s 2 3 1 3! +3 3 + 2 + s4 s s s 1 1 + s s−4 1 2 − * 20. {2t4 } = 2 kt {e −st (cos t)e 4! s5 19. 35. + −kt −e 7 3 + s2 s 2 16 9 + 2 + s3 s s 3! 2 6 1 − 12 3 + 2 − s4 s s s 1 2 5 − + s3 s+9 s . 1 1 k 1 }= − = 2 2 s−k s+k s − k2 s s2 − k 2 % / % / t −t 1 1 1 2t 1 t t e −e {e sinh t} = e = e − = − 2 2 2 2(s − 2) 2s % / % / t −t e +e 1 1 −2t 1 1 {e−t cosh t} = e−t = + e + = 2 2 2 2s 2(s + 2) / % 2 1 sin 4t = 2 {sin 2t cos 2t} = 2 s + 16 % / 1 1 1 1 s {cos2 t} = + cos 2t = + 2 2 2s 2 s2 + 4 {ekt + ekt } = 39. From the addition formula for the sine function, sin(4t + 5) = sin 4t cos 5 + cos 4t sin 5 so {sin(4t + 5)} = (cos 5) {sin 4t} + (sin 5) {cos 4t} = (cos 5) s2 4 s 4 cos 5 + (sin 5)s + (sin 5) 2 = . + 16 s + 16 s2 + 16 40. From the addition formula for the cosine function, √ 0 3 π π1 π 1 = cos t cos + sin t sin = cos t + sin t cos t − 6 6 6 2 2 so √ 2 0 3 1 π 13 = {cos t} + {sin t} cos t − 6 2 2 √ √ 1 1 1 3s + 1 3 s + = . = 2 s2 + 1 2 s2 + 1 2 s2 + 1 200 4.1 41. (a) Using integration by parts for α > 0, ! ! ∞ "∞ α −t α −t " t e dt = −t e " + α Γ(α + 1) = 0 0 (b) Let u = st so that du = s dt. Then ! ! ∞ −st α α {t } = e t dt = 42. (a) {t Γ(1/2) } = 1/2 = s 4 e−u 0 0 −1/2 ∞ π s tα−1 e−t dt = αΓ(α). 0 0 u 1α 1 1 du = α+1 Γ(α + 1), s s s {t1/2 } = (b) ∞ Definition of the Laplace Transform √ Γ(3/2) π = 3/2 3/2 s 2s 43. Let F (t) = t1/3 . Then F (t) is of exponential order, but f (t) = F ′ (t) = {t3/2 } = (c) 1 −2/3 3t α > −1. √ Γ(5/2) 3 π = s5/2 4s5/2 is unbounded near t = 0 and hence is not of exponential order. Let 2 2 f (t) = 2tet cos et = 2 d sin et . dt This function is not of exponential order, but we can show that its Laplace transform exists. Using integration by parts we have . $ # ! a ! ∞ "a d −st t2 t2 −st t2 t2 −st t2 " e sin e dt {2te cos e } = sin e e dt = lim e sin e " + s a→∞ dt 0 0 0 ! ∞ 2 2 e−st sin et dt = s {sin et } − sin 1. = − sin 1 + s 0 2 Since sin et is continuous and of exponential order, 2 {sin et } exists, and therefore 2 2 {2tet cos et } exists. 44. The relation will be valid when s is greater than the maximum of c1 and c2 . 2 45. Since et is an increasing function and t2 > ln M + ct for M > 0 we have et > eln M +ct = M ect for t sufficiently 2 large and for any c. Thus, et is not of exponential order. 46. Assuming that (c) of Theorem 4.1 is applicable with a complex exponent, we have {e(a+ib)t } = 1 s − a + ib (s − a) + ib 1 . = = s − (a + ib) (s − a) − ib (s − a) + ib (s − a)2 + b2 By Euler’s formula, eiθ = cos θ + i sin θ, so {e(a+ib)t } = = = {eat eibt } = {eat cos bt} + i {eat (cos bt + i sin bt)} {eat sin bt} s−a b +i . (s − a)2 + b2 (s − a)2 + b2 Equating real and imaginary parts we get {eat cos bt} = s−a (s − a)2 + b2 and {eat sin bt} = b . (s − a)2 + b2 47. We want f (αx + βy) = αf (x) + βf (y) or m(αx + βy) + b = α(mx + b) + β(my + b) = m(αx + βy) + (α + β)b for all real numbers α and β. Taking α = β = 1 we see that b = 2b, so b = 0. Thus, f (x) = mx + b will be a linear transformation when b = 0. 201 4.1 Definition of the Laplace Transform 48. Assume that parts, we have {tn−1 } = (n − 1)!/sn . Then, using the definition of the Laplace transform and integration by ∞ ! "∞ 1 n ∞ −st n−1 " e−st tn dt = − e−st tn " + e t dt s s 0 0 0 n! n (n − 1)! n = n+1 . {tn−1 } = =0+ s s sn s {tn } = ! EXERCISES 4.2 1. % 1 s3 2. % 1 s4 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. / = / = 1 2 % 2 s3 / = The Inverse Transform and Transforms of Derivatives 1 2 t 2 % / 3! 1 1 = t3 6 s4 6 % / % / 1 48 48 4! 1 − − = = t − 2t4 · s2 s5 s2 24 s5 5# % $2 6 / 1 1 4 3! 1 5! 1 5 2 2 = 4· 2 − · 4 + − 3 · 6 = 4t − t3 + t s s s 6 s 120 s 3 120 % / % / (s + 1)3 3 2 1 3! 1 1 3 1 + 3 · + · + · = = 1 + 3t + t2 + t3 4 2 3 4 s s s 2 s 6 s 2 6 % / % / (s + 2)2 2 1 1 + 4 · 2 + 2 · 3 = 1 + 4t + 2t2 = 3 s s s s / % 1 1 1 − + = t − 1 + e2t 2 s s s−2 % / % / 4 6 1 1 1 4! 1 1 + 5− = 4· + · 5 − = 4 + t4 − e−8t s s s+8 s 4 s s+8 4 % / % / 1 1 1 1 = = e−t/4 4s + 1 4 s + 1/4 4 % / % / 1 1 1 1 · = = e2t/5 5s − 2 5 s − 2/5 5 / % / % 5 5 7 5 = = sin 7t · s2 + 49 7 s2 + 49 7 % / 10s = 10 cos 4t s2 + 16 % / % / 4s 1 s = = cos t 4s2 + 1 s2 + 1/4 2 % / % / 1 1 1/2 1 1 = · = sin t 4s2 + 1 2 s2 + 1/4 2 2 202 4.2 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. % / s 3 − 2 · = 2 cos 3t − 2 sin 3t s2 + 9 s2 + 9 5 √ √ 6 % / √ √ s+1 1 s 2 2 = +√ 2 = cos 2t + sin 2 t s2 + 2 s2 + 2 s + 2 2 2 / % / % 1 1 1 1 1 1 1 = · − · = − e−3t s2 + 3s 3 s 3 s+3 3 3 % / % / 1 s+1 1 1 5 1 5 · + · = − = − + e4t 2 s − 4s 4 s 4 s−4 4 4 % / % / s 1 3 1 3 1 1 · + · = = et + e−3t 2 s + 2s − 3 4 s−1 4 s+3 4 4 / % / % 1 1 1 1 1 1 1 = = e4t − e−5t · − · s2 + s − 20 9 s−4 9 s+5 9 9 % / % / 0.9s 1 1 = (0.3) · + (0.6) · = 0.3e0.1t + 0.6e−0.2t (s − 0.1)(s + 0.2) s − 0.1 s + 0.2 5 √ 6 / % √ √ √ √ s 3 s−3 √ √ = cosh 3 t − 3 sinh 3 t − 3· 2 = 2 s −3 s −3 (s − 3 )(s + 3 ) % / % / 1 1 1 1 1 1 1 s · − + · = = e2t − e3t + e6t (s − 2)(s − 3)(s − 6) 2 s−2 s−3 2 s−6 2 2 % / % / s2 + 1 1 1 1 5 1 1 1 · − − · + · = s(s − 1)(s + 1)(s − 2) 2 s s−1 3 s+1 6 s−2 1 5 1 = − et − e−t + e2t 2 3 6 % / % / % / √ 1 1 1 1 1 s 1 1 5t = = · − = − cos s3 + 5s s(s2 + 5) 5 s 5 s2 + 5 5 5 % / % / s s 1 2 1 1 1 1 1 1 = · + · − · = cos 2t + sin 2t − e−2t (s2 + 4)(s + 2) 4 s2 + 4 4 s2 + 4 4 s + 2 4 4 4 % / % / % / 3 s 3 2s − 4 2s − 4 4 + + + = = − (s2 + s)(s2 + 1) s(s + 1)(s2 + 1) s s + 1 s2 + 1 s2 + 1 2s − 6 s2 + 9 / The Inverse Transform and Transforms of Derivatives = % 2· = −4 + 3e−t + cos t + 3 sin t 5 √ √ 6 % / √ √ 1 1 1 3 3 1 1 √ · 2 = √ sinh 3 t − √ sin 3 t − √ · 2 = 4 s −9 6 3 s −3 6 3 s +3 6 3 6 3 % / % / % / 1 1 1 1 1 2 1 1 1 · − · · − · = = (s2 + 1)(s2 + 4) 3 s2 + 1 3 s2 + 4 3 s2 + 1 6 s2 + 4 1 1 = sin t − sin 2t 3 6 % / % / 6s + 3 s 1 s 1 2 = 2· 2 + −2· 2 − · (s2 + 1)(s2 + 4) s + 1 s2 + 1 s + 4 2 s2 + 4 1 = 2 cos t + sin t − 2 cos 2t − sin 2t 2 The Laplace transform of the initial-value problem is 1 s {y} − y(0) − {y} = . s 203 4.2 The Inverse Transform and Transforms of Derivatives Solving for {y} we obtain 1 1 {y} = − + . s s−1 Thus y = −1 + et . 32. The Laplace transform of the initial-value problem is 2s Solving for {y} − 2y(0) + {y} = 0. {y} we obtain 6 3 = . 2s + 1 s + 1/2 {y} = Thus y = 3e−t/2 . 33. The Laplace transform of the initial-value problem is s Solving for {y} − y(0) + 6 {y} = 1 . s−4 {y} we obtain {y} = 1 2 1 1 19 1 + = · + · . (s − 4)(s + 6) s + 6 10 s − 4 10 s + 6 Thus y= 1 4t 19 −6t e + e . 10 10 34. The Laplace transform of the initial-value problem is s Solving for {y} − {y} = s2 2s . + 25 {y} we obtain {y} = 1 1 1 5 5 s 2s = · − + · 2 . 2 2 (s − 1)(s + 25) 13 s − 1 13 s + 25 13 s + 25 Thus y= 1 t 1 5 e − cos 5t + sin 5t. 13 13 13 35. The Laplace transform of the initial-value problem is s2 Solving for {y} − sy(0) − y ′ (0) + 5 [s {y} − y(0)] + 4 {y} = 0. {y} we obtain {y} = 4 1 1 1 s+5 = − . s2 + 5s + 4 3 s+1 3 s+4 Thus y= 4 −t 1 −4t e − e . 3 3 36. The Laplace transform of the initial-value problem is s2 {y} − sy(0) − y ′ (0) − 4 [s {y} − y(0)] = 204 6 3 − . s−3 s+1 4.2 Solving for The Inverse Transform and Transforms of Derivatives {y} we obtain {y} = = Thus 6 3 s−5 − + 2 2 2 (s − 3)(s − 4s) (s + 1)(s − 4s) s − 4s 2 3 1 11 1 5 1 · − − · + · . 2 s s − 3 5 s + 1 10 s − 4 y= 5 3 11 − 2e3t − e−t + e4t . 2 5 10 37. The Laplace transform of the initial-value problem is s2 Solving for {y} − sy(0) + {y} = 2 . s2 + 2 {y} we obtain {y} = 10s 10s 2 2 2 + = 2 + − . (s2 + 1)(s2 + 2) s2 + 1 s + 1 s2 + 1 s2 + 2 Thus y = 10 cos t + 2 sin t − √ √ 2 sin 2 t. 38. The Laplace transform of the initial-value problem is s2 Solving for {y} + 9 {y} = {y} we obtain {y} = 1 . s−1 1 1 1 1 1 1 s = · − · 2 − · 2 . 2 (s − 1)(s + 9) 10 s − 1 10 s + 9 10 s + 9 Thus y= 1 t 1 1 e − sin 3t − cos 3t. 10 30 10 39. The Laplace transform of the initial-value problem is 7 2 s3 8 7 {y} − s2 (0) − sy ′ (0) − y ′′ (0) + 3 s2 Solving for {y} we obtain {y} = 8 {y} − sy(0) − y ′ (0) − 3[s {y} − y(0)] − 2 {y} = 1 . s+1 1 1 5 8 1 1 1 1 2s + 3 = + − + . (s + 1)(s − 1)(2s + 1)(s + 2) 2 s + 1 18 s − 1 9 s + 1/2 9 s + 2 Thus y= 1 −t 5 8 1 e + et − e−t/2 + e−2t . 2 18 9 9 40. The Laplace transform of the initial-value problem is s3 7 {y} − s2 (0) − sy ′ (0) − y ′′ (0) + 2 s2 Solving for {y} we obtain {y} = = 8 {y} − sy(0) − y ′ (0) − [s {y} − y(0)] − 2 s2 + 12 (s − 1)(s + 1)(s + 2)(s2 + 9) 13 1 13 1 16 1 3 1 s 3 − + + − . 60 s − 1 20 s + 1 39 s + 2 130 s2 + 9 65 s2 + 9 205 {y} = 3 . s2 + 9 4.2 The Inverse Transform and Transforms of Derivatives Thus y= 13 t 13 −t 16 −2t 3 1 e − e + e + cos 3t − sin 3t. 60 20 39 130 65 41. The Laplace transform of the initial-value problem is s Solving for {y} + {y} = s+3 . s2 + 6s + 13 {y} we obtain 1 1 1 s+1 s+3 = · − · 2 2 (s + 1)(s + 6s + 13) 4 s + 1 4 s + 6s + 13 # $ 1 1 2 s+3 1 − − . = · 4 s + 1 4 (s + 3)2 + 4 (s + 3)2 + 4 {y} = Thus y= 1 −t 1 −3t 1 e − e cos 2t + e−3t sin 2t. 4 4 4 42. The Laplace transform of the initial-value problem is s2 Solving for {y} − s · 1 − 3 − 2[s {y} − 1] + 5 {y} = (s2 − 2s + 5) {y} − s − 1 = 0. {y} we obtain {y} = s+1 s−1+2 s−1 2 = = + . s2 − 2s + 5 (s − 1)2 + 22 (s − 1)2 + 22 (s − 1)2 + 22 Thus y = et cos 2t + et sin 2t. 43. (a) Differentiating f (t) = teat we get f ′ (t) = ateat + eat so f (0) = 0. Writing the equation as {teat } + a and solving for {ateat + eat } = s {eat } = s {teat }, where we have used {teat } {teat } we get {teat } = 1 s−a {eat } = 1 . (s − a)2 (b) Starting with f (t) = t sin kt we have f ′ (t) = kt cos kt + sin kt f ′′ (t) = −k 2 t sin kt + 2k cos kt. Then {−k 2 t sin t + 2k cos kt} = s2 {t sin kt} where we have used f (0) = 0 and f ′ (0) = 0. Writing the above equation as −k 2 and solving for {t sin kt} + 2k {cos kt} = s2 {t sin kt} {t sin kt} gives {t sin kt} = 44. Let f1 (t) = 1 and f2 (t) = % 1, 0, 2k s2 + k 2 {cos kt} = t ≥ 0, t ̸= 1 . Then t=1 2k s 2ks = 2 . s2 + k 2 s2 + k 2 (s + k 2 )2 {f1 (t)} = 206 {f2 (t)} = 1/s, but f1 (t) ̸= f2 (t). 4.3 Translation Theorems 45. For y ′′ − 4y ′ = 6e3t − 3e−t the transfer function is W (s) = 1/(s2 − 4s). The zero-input response is % / % / s−5 5 1 1 5 1 1 y0 (t) = = = − e4t , · − · s2 − 4s 4 s 4 s−4 4 4 and the zero-state response is y1 (t) = 3 6 − (s − 3)(s2 − 4s) (s + 1)(s2 − 4s) = % 27 1 2 5 1 3 1 · − + · − · 20 s − 4 s − 3 4 s 5 s + 1 = / 27 4t 5 3 e − 2e3t + − e−t . 20 4 5 46. From Theorem 4.4, if f and f ′ are continuous and of exponential order, {f ′ (t)} = sF (s) − f (0). From {f ′ (t)} = 0 so Theorem 4.5, lims→∞ / % lim [sF (s) − f (0)] = 0 and s→∞ For f (t) = cos kt, lim sF (s) = lim s s→∞ s→∞ s2 lim F (s) = f (0). s→∞ s = 1 = f (0). + k2 EXERCISES 4.3 Translation Theorems 1 (s − 10)2 2. 9 : te−6t = 3! (s + 2)4 4. 9 : t10 e−7t = 1. 9 : te10t = 3. 9 : t3 e−2t = 5. 2 ; <2 3 t et + e2t = 6. 9 : e2t (t − 1)2 = 7. 9 : et sin 3t = 9. {(1 − et + 3e−4t ) cos 5t} = 10. 11. 9 9 : te2t + 2te3t + te4t = : t2 e2t − 2te2t + e2t = 3 (s − 1)2 + 9 1 (s + 6)2 10! (s + 7)11 2 1 1 + + 2 2 (s − 2) (s − 3) (s − 4)2 2 1 2 − + (s − 2)3 (s − 2)2 s−2 8. 9 : e−2t cos 4t = {cos 5t − et cos 5t + 3e−4t cos 5t} = s+2 (s + 2)2 + 16 s−1 3(s + 4) s − + s2 + 25 (s − 1)2 + 25 (s + 4)2 + 25 % # $/ % / t t 5 9 4 e3t 9 − 4t + 10 sin = 9e3t − 4te3t + 10e3t sin = + − 2 2 s − 3 (s − 3)2 (s − 3)2 + 1/4 % / % / 1 1 2 1 = = t2 e−2t (s + 2)3 2 (s + 2)3 2 207 4.3 Translation Theorems 12. % 1 (s − 1)4 13. % s2 / = % 1 6 / 3! (s − 1)4 % / = 1 3 t t e 6 / 1 1 = e3t sin t = − 6s + 10 (s − 3)2 + 12 / / % % 1 2 1 1 = e−t sin 2t = 2 2 2 s + 2s + 5 2 (s + 1) + 2 2 % / / % s 1 s+2 −2 = e−2t cos t − 2e−2t sin t = s2 + 4s + 5 (s + 2)2 + 12 (s + 2)2 + 12 / / % % 1 1 2s + 5 5 (s + 3) − = 2e−3t cos 5t − e−3t sin 5t = 2 2 2 2 2 2 s + 6s + 34 (s + 3) + 5 5 (s + 3) + 5 5 / % / % / % s+1−1 1 1 s = = = e−t − te−t − (s + 1)2 (s + 1)2 s + 1 (s + 1)2 % / % / % / 5s 10 5(s − 2) + 10 5 + = = = 5e2t + 10te2t (s − 2)2 (s − 2)2 s − 2 (s − 2)2 / % / % 1 5 4 3 5 2 3 2s − 1 − − = = 5 − t − 5e−t − 4te−t − t2 e−t − − s2 (s + 1)3 s s2 s + 1 (s + 1)2 2 (s + 1)3 2 % / % / (s + 1)2 2 1 1 3! 1 − + = = te−2t − t2 e−2t + t3 e−2t (s + 2)4 (s + 2)2 (s + 2)3 6 (s + 2)4 6 14. 15. 16. 17. 18. 19. 20. 21. The Laplace transform of the differential equation is s Solving for {y} − y(0) + 4 {y} = 1 . s+4 {y} we obtain {y} = 1 2 + . 2 (s + 4) s+4 Thus y = te−4t + 2e−4t . 22. The Laplace transform of the differential equation is s Solving for {y} − {y} = 1 1 + . s (s − 1)2 {y} we obtain {y} = Thus 1 1 1 1 1 + =− + + . 3 s(s − 1) (s − 1) s s − 1 (s − 1)3 1 y = −1 + et + t2 et . 2 23. The Laplace transform of the differential equation is s2 Solving for {y} we obtain 7 {y} − sy(0) − y ′ (0) + 2 s {y} = 8 {y} − y(0) + s+3 1 2 = + . (s + 1)2 s + 1 (s + 1)2 Thus y = e−t + 2te−t . 208 {y} = 0. 4.3 Translation Theorems 24. The Laplace transform of the differential equation is s2 {y} − sy(0) − y ′ (0) − 4 [s {y} − y(0)] + 4 {y} = 6 . (s − 2)4 1 1 5 2t 5! . Thus, y = t e . 20 (s − 2)6 20 25. The Laplace transform of the differential equation is Solving for {y} we obtain s2 Solving for {y} = {y} − sy(0) − y ′ (0) − 6 [s {y} − y(0)] + 9 {y} = 1 . s2 {y} we obtain {y} = 1 + s2 2 1 1 1 2 10 1 1 = + − + . 2 2 2 s (s − 3) 27 s 9 s 27 s − 3 9 (s − 3)2 Thus y= 2 1 2 10 + t − e3t + te3t . 27 9 27 9 26. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) − 4 [s {y} − y(0)] + 4 {y} = 6 . s4 {y} we obtain {y} = s5 − 4s4 + 6 3 1 9 1 3 2 1 3! 1 1 13 1 = + + + . + − s4 (s − 2)2 4 s 8 s2 4 s3 4 s4 4 s−2 8 (s − 2)2 Thus y= 3 9 1 1 13 3 + t + t2 + t3 + e2t − te2t . 4 8 4 4 4 8 27. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) − 6 [s {y} = 0. {y} we obtain {y} = − Thus 3 2 3 =− . s2 − 6s + 13 2 (s − 3)2 + 22 3 y = − e3t sin 2t. 2 28. The Laplace transform of the differential equation is 7 8 7 2 s2 {y} − sy(0) + 20 s Solving for {y} − y(0)] + 13 {y} we obtain {y} = 2s2 Thus 8 {y} − y(0) + 51 {y} = 0. 4s + 40 2s + 20 2(s + 5) 10 = = + . 2 2 + 20s + 51 (s + 5) + 1/2 (s + 5) + 1/2 (s + 5)2 + 1/2 √ √ √ y = 2e−5t cos(t/ 2 ) + 10 2 e−5t sin(t/ 2 ). 29. The Laplace transform of the differential equation is s2 {y} − sy(0) − y ′ (0) − [s {y} − y(0)] = 209 s−1 . (s − 1)2 + 1 4.3 Translation Theorems Solving for {y} we obtain {y} = s(s2 1 1 1 1 1 s−1 1 = − + . 2 − 2s + 2) 2 s 2 (s − 1) + 1 2 (s − 1)2 + 1 Thus 1 1 1 t − e cos t + et sin t. 2 2 2 y= 30. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) − 2 [s {y} − y(0)] + 5 {y} = 1 1 . + s s2 {y} we obtain {y} = = 4s2 + s + 1 −7s/25 + 109/25 7 1 1 1 + = + s2 (s2 − 2s + 5) 25 s 5 s2 s2 − 2s + 5 7 51 s−1 2 7 1 1 1 − + . + 25 s 5 s2 25 (s − 1)2 + 22 25 (s − 1)2 + 22 Thus y= 7 1 7 51 + t − et cos 2t + et sin 2t. 25 5 25 25 31. Taking the Laplace transform of both sides of the differential equation and letting c = y(0) we obtain {y ′′ } + 2 ′ s {y} − sy(0) − y (0) + 2s s2 {2y ′ } + {y} = 0 {y} − 2y(0) + {y} = 0 {y} − cs − 2 + 2s ; {y} − 2c + < s2 + 2s + 1 {y} = 0 {y} = cs + 2c + 2 cs 2c + 2 {y} = + 2 (s + 1) (s + 1)2 =c = 2c + 2 s+1−1 + (s + 1)2 (s + 1)2 c c+2 . + s + 1 (s + 1)2 Therefore, % y(t) = c 1 s+1 / % + (c + 2) 1 (s + 1)2 / = ce−t + (c + 2)te−t . To find c we let y(1) = 2. Then 2 = ce−1 + (c + 2)e−1 = 2(c + 1)e−1 and c = e − 1. Thus y(t) = (e − 1)e−t + (e + 1)te−t . 32. Taking the Laplace transform of both sides of the differential equation and letting c = y ′ (0) we obtain {y ′′ } + 2 s {8y ′ } + ′ {20y} = 0 {y} − y (0) + 8s {y} + 20 {y} = 0 {y} − c + 8s {y} + 20 {y} = 0 (s + 8s + 20) {y} = c s2 2 {y} = 210 c c = . s2 + 8s + 20 (s + 4)2 + 4 4.3 Translation Theorems Therefore, % y(t) = c (s + 4)2 + 4 / = c −4t e sin 2t = c1 e−4t sin 2t. 2 To find c we let y ′ (π) = 0. Then 0 = y ′ (π) = ce−4π and c = 0. Thus, y(t) = 0. (Since the differential equation is homogeneous and both boundary conditions are 0, we can see immediately that y(t) = 0 is a solution. We have shown that it is the only solution.) 33. Recall from Section 3.8 that mx′′ = −kx − βx′ . Now m = W/g = 4/32 = 81 slug, and 4 = 2k so that k = 2 lb/ft. Thus, the differential equation is x′′ + 7x′ + 16x = 0. The initial conditions are x(0) = −3/2 and x′ (0) = 0. The Laplace transform of the differential equation is s2 Solving for 3 {x} + s + 7s 2 {x} + 21 + 16 2 {x} = 0. {x} we obtain √ √ −3s/2 − 21/2 7 15 15/2 3 s + 7/2 √ √ {x} = 2 − . =− s + 7s + 16 2 (s + 7/2)2 + ( 15/2)2 10 (s + 7/2)2 + ( 15/2)2 Thus √ √ √ 3 −7t/2 7 15 −7t/2 15 15 x=− e cos t− e sin t. 2 2 10 2 34. The differential equation is dq d2 q + 20 + 200q = 150, 2 dt dt The Laplace transform of this equation is s2 Solving for {q} + 20s q(0) = q ′ (0) = 0. {q} + 200 150 . s {q} = {q} we obtain {q} = s(s2 150 3 1 3 3 s + 10 10 = − − . 2 2 + 20s + 200) 4 s 4 (s + 10) + 10 4 (s + 10)2 + 102 Thus q(t) = 3 3 3 −10t − e cos 10t − e−10t sin 10t 4 4 4 and i(t) = q ′ (t) = 15e−10t sin 10t. 35. The differential equation is d2 q dq E0 + 2λ + ω 2 q = , 2 dt dt L The Laplace transform of this equation is s2 {q} + 2λs {q} + ω 2 q(0) = q ′ (0) = 0. {q} = or Solving for ; s2 + 2λs + ω 2 < {q} = E0 1 L s E0 1 . L s {q} and using partial fractions we obtain $ # $ # (1/ω 2 )s + 2λ/ω 2 s + 2λ E0 1 E0 1/ω 2 − 2 − = . {q} = L s s + 2λs + ω 2 Lω 2 s s2 + 2λs + ω 2 211 4.3 Translation Theorems < ; For λ > ω we write s2 + 2λs + ω 2 = (s + λ)2 − λ2 − ω 2 , so (recalling that ω 2 = 1/LC) # $ 1 s+λ λ {q} = E0 C − − . s (s + λ)2 − (λ2 − ω 2 ) (s + λ)2 − (λ2 − ω 2 ) Thus for λ > ω, $. # = = λ sinh λ2 − ω 2 t . q(t) = E0 C 1 − e−λt cosh λ2 − ω 2 t − √ λ2 − ω 2 ; < For λ < ω we write s2 + 2λs + ω 2 = (s + λ)2 + ω 2 − λ2 , so # $ 1 s+λ λ {q} = E0 C − − . s (s + λ)2 + (ω 2 − λ2 ) (s + λ)2 + (ω 2 − λ2 ) Thus for λ < ω, $. # = = λ sin ω 2 − λ2 t . q(t) = E0 C 1 − e−λt cos ω 2 − λ2 t − √ ω 2 − λ2 For λ = ω, s2 + 2λ + ω 2 = (s + λ)2 and $ # $ # E0 1/λ2 1/λ2 1/λ 1 λ 1 E0 1 E0 = − − − − = . {q} = L s(s + λ)2 L s s + λ (s + λ)2 Lλ2 s s + λ (s + λ)2 Thus for λ = ω, ; < q(t) = E0 C 1 − e−λt − λte−λt . 36. The differential equation is dq 1 + q = E0 e−kt , q(0) = 0. dt C The Laplace transform of this equation is R Rs Solving for {q} + 1 C {q} = E0 1 . s+k {q} we obtain {q} = E0 C E0 /R = . (s + k)(RCs + 1) (s + k)(s + 1/RC) When 1/RC ̸= k we have by partial fractions # $ # $ E0 1/(1/RC − k) 1/(1/RC − k) 1 E0 1 1 {q} = − = − . R s+k s + 1/RC R 1/RC − k s + k s + 1/RC Thus q(t) = When 1/RC = k we have 1 E0 C 0 −kt e − e−t/RC . 1 − kRC {q} = Thus q(t) = 37. 9 38. 9 (t − 1) e2−t : e−s (t − 1) = 2 s 2 : (t − 2) = e−(t−2) E0 1 . R (s + k)2 E0 −t/RC E0 −kt = . te te R R 3 e−2s (t − 2) = s+1 212 4.3 39. 9 t : (t − 2) = {(t − 2) (t − 2) + 2 (t − 2)} = Alternatively, (16) of this section could be used: {t 40. 9 : (t − 1) = 3 (3t + 1) 9 −2s (t − 1) {t + 2} = e : (t − 1) + 4 41. 9 : (t − π) = cos 2t {cos 2(t − π) (t − 1)} = e−s (t − π)} = # 2 1 + 2 s s $ . : 3e−s 4e−s (t − 1) = 2 + s s 9 Alternatively, (16) of this section could be used: {(3t + 1) e−2s 2e−2s + 2 s s −2s (t − 2)} = e Translation Theorems {3t + 4} = e−s # 4 3 + s2 s $ . se−πs s2 + 4 Alternatively, (16) of this section could be used: {cos 2t 42. 2 sin t 0 t− π 13 = 2 (t − π)} = e−πs 2 0 π1 cos t − 2 0 {cos 2(t + π)} = e−πs t− 44. 45. 46. 47. 48. e−2s s3 49. (c) 57. {cos t} = e−πs/2 s . s2 + 1 / 1 2 −2s 1 · 3e = (t − 2)2 (t − 2) 2 s 2 % / % / (1 + e−2s )2 2e−2s e−4s 1 + + = = e−2t + 2e−2(t−2) (t − 2) + e−2(t−4) (t − 4) s+2 s+2 s+2 s+2 % −πs / e = sin(t − π) (t − π) = − sin t (t − π) s2 + 1 % −πs/2 / 0 0 se π1 0 π1 π1 = cos 2 t − t − = − cos 2t t − s2 + 4 2 2 2 / % / % e−s e−s e−s = − = (t − 1) − e−(t−1) (t − 1) s(s + 1) s s+1 % / % −2s / e−2s e e−2s e−2s = − − 2 + = − (t − 2) − (t − 2) (t − 2) + et−2 (t − 2) s2 (s − 1) s s s−1 % 43. 56. s . s2 + 4 π 13 se−πs/2 = 2 2 s +1 Alternatively, (16) of this section could be used: 2 0 2 0 π 13 π 13 sin t t− = e−πs/2 = e−πs/2 sin t + 2 2 55. {cos 2t} = e−πs / = 50. (e) % 51. (f ) 52. (b) : 2 4 (t − 3) = − e−3s s s 9 : 1 e−4s e−5s 1 − (t − 4) + (t − 5) = − + s s s 92 : 97 8 : t (t − 1) = (t − 1)2 + 2t − 1 (t − 1) = $ # 2 1 −s 2 + + e = s3 s2 s 9 53. (a) 54. (d) 2−4 213 97 8 (t − 1)2 + 2(t − 1) − 1 : (t − 1) 4.3 Translation Theorems Alternatively, by (16) of this section, {t2 58. 59. # 9 : (t − 2) = t−t t− 60. 9 sin t − sin t 61. 9 : f (t) = 62. $/ % sin t 9 : f (t) = 3π 2 % # $ 3π − cos t − 2 = 9 9 {t2 + 2t + 1} = e−s # t− 3π 2 $/ =− # 2 2 1 + 2+ s3 s s $ . se−3πs/2 s2 + 1 : 1 e−2s 2e−2s (t − 2) = 2 − 2 − s s s −2πs 9 : e 1 − sin t − sin(t − 2π) (t − 2π) = 2 s + 1 s2 + 1 t − (t − 2) : (t − 2π) = 9 (t − 1)} = e−s (t − 2) − 2 (t − a) − : e−as e−bs (t − b) = − s s (t − 1) + (t − 2) + : e−s e−2s e−3s 1 e−s (t − 3) + · · · = + + + ··· = s s s s 1 − e−s 63. The Laplace transform of the differential equation is s Solving for {y} − y(0) + {y} = 5 −s e . s {y} we obtain . 5e−s 1 −s 1 {y} = = 5e − . s(s + 1) s s+1 Thus y=5 (t − 1) − 5e−(t−1) (t − 1). 64. The Laplace transform of the differential equation is s Solving for {y} − y(0) + {y} = 1 2 −s − e . s s {y} we obtain . 1 2e−s 1 1 1 −s 1 {y} = − = − − 2e − . s(s + 1) s(s + 1) s s+1 s s+1 Thus > ? y = 1 − e−t − 2 1 − e−(t−1) (t − 1). 65. The Laplace transform of the differential equation is s Solving for {y} = s+1 1 − e−s 2 . 2 s s {y} we obtain {y} = Thus {y} − y(0) + 2 . 1 1 1 1 1 1 1 1 1 1 1 −s s + 1 −s 1 1 + − . − e = − + − e + s2 (s + 2) s2 (s + 2) 4 s 2 s2 4 s+2 4 s 2 s2 4 s+2 . 1 1 1 1 1 1 y = − + t + e−2t − + (t − 1) − e−2(t−1) (t − 1). 4 2 4 4 2 4 66. The Laplace transform of the differential equation is s2 {y} − sy(0) − y ′ (0) + 4 214 {y} = 1 e−s − . s s 4.3 Solving for Translation Theorems {y} we obtain {y} = . 1−s 1 1 1 1 s 1 2 1 s −s −s 1 1 − e = − − − e − . s(s2 + 4) s(s2 + 4) 4 s 4 s2 + 4 2 s2 + 4 4 s 4 s2 + 4 Thus y= . 1 1 1 1 1 − cos 2t − sin 2t − − cos 2(t − 1) (t − 1). 4 4 2 4 4 67. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) + 4 {y} = e−2πs 1 . s2 + 1 {y} we obtain . 1 2 1 s −2πs 1 + e − . s2 + 4 3 s2 + 1 6 s2 + 4 {y} = Thus y = cos 2t + - . 1 1 sin(t − 2π) − sin 2(t − 2π) (t − 2π). 3 6 68. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) − 5 [s {y} − y(0)] + 6 {y} = e−s . s {y} we obtain 1 1 + s(s − 2)(s − 3) (s − 2)(s − 3) . 1 1 1 1 1 1 −s 1 1 =e − + − + . 6 s 2 s−2 3 s−3 s−2 s−3 {y} = e−s Thus - 1 1 2(t−1) 1 3(t−1) − e + e y= 6 2 3 . (t − 1) − e2t + e3t . 69. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) + {y} = e−πs e−2πs − . s s {y} we obtain −πs {y} = e - . . 1 s s 1 −2πs 1 − −e − + 2 . s s2 + 1 s s2 + 1 s +1 Thus y = [1 − cos(t − π)] (t − π) − [1 − cos(t − 2π)] (t − 2π) + sin t. 70. The Laplace transform of the differential equation is s2 Solving for 7 {y} − sy(0) − y ′ (0) + 4 s {y} we obtain 8 {y} − y(0) + 3 {y} = 1 e−2s e−4s e−6s − − + . s s s s . 1 1 1 1 1 1 1 1 1 1 −2s 1 1 − + −e − + {y} = 3 s 2 s+1 6 s+3 3 s 2 s+1 6 s+3 . . 1 1 1 1 1 1 1 1 1 1 1 1 − e−4s − + + e−6s − + . 3 s 2 s+1 6 s+3 3 s 2 s+1 6 s+3 215 4.3 Translation Theorems Thus y= . 1 1 −t 1 −3t 1 1 −(t−2) 1 −3(t−2) − + e (t − 2) − e + e − e 3 2 6 3 2 6 . . 1 1 −(t−6) 1 −3(t−6) 1 1 −(t−4) 1 −3(t−4) − e + e − e + e (t − 4) + (t − 6). − 3 2 6 3 2 6 71. Recall from Section 3.8 that mx′′ = −kx + f (t). Now m = W/g = 32/32 = 1 slug, and 32 = 2k so that k = 16 lb/ft. Thus, the differential equation is x′′ + 16x = f (t). The initial conditions are x(0) = 0, x′ (0) = 0. Also, since 5 20t, 0 ≤ t < 5 f (t) = 0, t≥5 and 20t = 20(t − 5) + 100 we can write f (t) = 20t − 20t (t − 5) = 20t − 20(t − 5) (t − 5) − 100 (t − 5). The Laplace transform of the differential equation is s2 Solving for {x} + 16 {x} = 20 20 −5s 100 −5s − 2e − e . s2 s s {x} we obtain 20 100 20 − e−5s − e−5s s2 (s2 + 16) s2 (s2 + 16) s(s2 + 16) $ $ # # ; < 5 4 s 25 1 25 5 1 −5s · − · · − · e−5s . 1−e − = 4 s2 16 s2 + 16 4 s 4 s2 + 16 {x} = Thus . . 5 5 25 25 5 5 x(t) = t − sin 4t − (t − 5) − sin 4(t − 5) (t − 5) − − cos 4(t − 5) (t − 5) 4 16 4 16 4 4 = 5 5 5 t− sin 4t − t 4 16 4 (t − 5) + 5 sin 4(t − 5) 16 (t − 5) + 25 cos 4(t − 5) 4 (t − 5). 72. Recall from Section 3.8 that mx′′ = −kx + f (t). Now m = W/g = 32/32 = 1 slug, and 32 = 2k so that k = 16 lb/ft. Thus, the differential equation is x′′ + 16x = f (t). The initial conditions are x(0) = 0, x′ (0) = 0. Also, since f (t) = and sin t = sin(t − 2π) we can write % sin t, 0 ≤ t < 2π 0, t ≥ 2π f (t) = sin t − sin(t − 2π) (t − 2π). The Laplace transform of the differential equation is s2 Solving for {x} + 16 {x} = 1 1 − e−2πs . s2 + 1 s2 + 1 {x} we obtain 1 1 − e−2πs (s2 + 16) (s2 + 1) (s2 + 16) (s2 + 1) . −1/15 −1/15 1/15 1/15 = 2 + 2 − 2 + 2 e−2πs . s + 16 s + 1 s + 16 s + 1 {x} = 216 4.3 Thus x(t) = − = 5 1 1 1 sin 4t + sin t + sin 4(t − 2π) 60 15 60 1 sin 4t + − 60 1 15 sin t, 0, (t − 2π) − 1 sin(t − 2π) 15 Translation Theorems (t − 2π) 0 ≤ t < 2π t ≥ 2π. 73. The differential equation is 2.5 dq + 12.5q = 5 dt (t − 3). {q} + 5 2 −3s e . s The Laplace transform of this equation is s Solving for {q} = {q} we obtain 2 e−3s = s(s + 5) {q} = Thus q(t) = 2 5 # 2 1 2 1 · − · 5 s 5 s+5 2 (t − 3) − e−5(t−3) 5 $ e−3s . (t − 3). 74. The differential equation is dq + 10q = 30et − 30et (t − 1.5). dt The Laplace transform of this equation is 10 s Solving for {q} − q0 + {q} = {q} we obtain {q} = Thus q(t) = # # 3e1.5 −1.5s 3 . − e s − 1 s − 1.5 q0 − 3 2 $ 3 2 $ 0 1 6 3 e−t + et + e1.5 e−(t−1.5) − e1.5(t−1.5) 2 5 q0 − · 1 3 1 + · − 3e1.5 s+1 2 s−1 # −2/5 2/5 + s+1 s − 1.5 $ e−1.5s . (t − 1.5). 75. (a) The differential equation is $ # $ # di 3π 3π t− , + 10i = sin t + cos t − dt 2 2 i(0) = 0. The Laplace transform of this equation is s Solving for {i} + 10 {i} = 1 se−3πs/2 + . s2 + 1 s2 + 1 {i} we obtain s 1 + e−3πs/2 (s2 + 1)(s + 10) (s2 + 1)(s + 10) # $ # $ s 10 10s 1 1 −10 1 1 − + + + + e−3πs/2 . = 101 s + 10 s2 + 1 s2 + 1 101 s + 10 s2 + 1 s2 + 1 {i} = Thus i(t) = < 1 ; −10t e − cos t + 10 sin t 101 # $ # $. # $ 1 3π 3π 3π + −10e−10(t−3π/2) + 10 cos t − + sin t − t− . 101 2 2 2 217 4.3 Translation Theorems i (b) 0.2 1 2 5 4 3 t 6 -0.2 The maximum value of i(t) is approximately 0.1 at t = 1.7, the minimum is approximately −0.1 at 4.7. 76. (a) The differential equation is dq 1 + q = E0 [ dt 0.01 (t − 1) − (t − 3)], q(0) = 0 dq + 100q = E0 [ dt The Laplace transform of this equation is (t − 1) − (t − 3)], q(0) = 0. 50 or 50 50s {q} + 100 {q} = E0 # $ 1 −s 1 −3s e − e . s s Solving for {q} we obtain . - # $ # $ . E0 e−s E0 1 1 1 1 e−3s 1 1 −s −3s {q} = = e − e − − − . 50 s(s + 2) s(s + 2) 50 2 s s + 2 2 s s+2 Thus q(t) = (b) q 1 0 1 ? E0 >0 1 − e−2(t−1) (t − 1) − 1 − e−2(t−3) (t − 3) . 100 1 1 2 4 3 5 6 t The maximum value of q(t) is approximately 1 at t = 3. 77. The differential equation is d4 y = w0 [1 − (x − L/2)]. dx4 Taking the Laplace transform of both sides and using y(0) = y ′ (0) = 0 we obtain 1 w0 1 0 1 − e−Ls/2 . s4 {y} − sy ′′ (0) − y ′′′ (0) = EI s Letting y ′′ (0) = c1 and y ′′′ (0) = c2 we have 1 c1 c2 w0 1 0 −Ls/2 {y} = 3 + 4 + 1 − e s s EI s5 so that ) # $4 # $, 1 1 1 w0 L L 2 3 4 y(x) = c1 x + c2 x + . x − x− x− 2 6 24 EI 2 2 EI To find c1 and c2 we compute ) # $2 1 w0 L y (x) = c1 + c2 x + x2 − x − 2 EI 2 ′′ and 218 # L x− 2 $, 4.3 y ′′′ (x) = c2 + Translation Theorems # $ # $. L L w0 x− x− x− . EI 2 2 Then y ′′ (L) = y ′′′ (L) = 0 yields the system ) # $2 , 1 w0 3 w0 L2 L c1 + c2 L + = c1 + c2 L + L2 − =0 2 EI 2 8 EI w0 c2 + EI # $ L 1 w0 L = c2 + = 0. 2 2 EI Solving for c1 and c2 we obtain c1 = 18 w0 L2 /EI and c2 = − 21 w0 L/EI. Thus ) # $4 # $, L 1 1 4 1 L w0 1 2 2 3 x− L x − Lx + x − x− . y(x) = EI 16 12 24 24 2 2 78. The differential equation is d4 y = w0 [ (x − L/3) − (x − 2L/3)]. dx4 Taking the Laplace transform of both sides and using y(0) = y ′ (0) = 0 we obtain 1 w0 1 0 −Ls/3 s4 {y} − sy ′′ (0) − y ′′′ (0) = e − e−2Ls/3 . EI s EI Letting y ′′ (0) = c1 and y ′′′ (0) = c2 we have {y} = so that 1 c1 c2 w0 1 0 −Ls/3 −2Ls/3 e − e + + s3 s4 EI s5 1 1 w0 1 y(x) = c1 x2 + c2 x3 + 2 6 24 EI )# L x− 3 $4 # L x− 3 $ # 2L − x− 3 $4 # 2L x− 3 $, . To find c1 and c2 we compute 1 w0 y (x) = c1 + c2 x + 2 EI ′′ and w0 y (x) = c2 + EI ′′′ )# L x− 3 $2 # L x− 3 $ # 2L − x− 3 $2 # 2L x− 3 $, -# $ # $ # $ # $. L L 2L 2L x− x− − x− x− . 3 3 3 3 Then y ′′ (L) = y ′′′ (L) = 0 yields the system )# $ # $2 , 2 1 w0 2L 1 w0 L2 L c1 + c2 L + =0 = c1 + c2 L + − 2 EI 3 3 6 EI c2 + w0 EI - . 2L L 1 w0 L − = c2 + = 0. 3 3 3 EI Solving for c1 and c2 we obtain c1 = 16 w0 L2 /EI and c2 = − 31 w0 L/EI. Thus * )# $4 # $ # $4 1 1 1 2 2 L L 2L w0 3 L x − Lx + x− x− − x− y(x) = EI 12 18 24 3 3 3 79. The differential equation is EI # $ d4 y 2w0 L L = − x + x − dx4 L 2 2 219 # x− L 2 $. . # 2L x− 3 $,+ . 4.3 Translation Theorems Taking the Laplace transform of both sides and using y(0) = y ′ (0) = 0 we obtain . 1 1 2w0 L s4 {y} − sy ′′ (0) − y ′′′ (0) = − 2 + 2 e−Ls/2 . EIL 2s s s Letting y ′′ (0) = c1 and y ′′′ (0) = c2 we have {y} = so that . c2 2w0 1 1 −Ls/2 c1 L + + − + e s3 s4 EIL 2s5 s6 s6 ) # $5 # $, 1 2w0 L 4 1 5 1 L 1 L 2 3 y(x) = c1 x + c2 x + x − x + x− x− 2 6 EIL 48 120 120 2 2 ) # $5 # $, 1 1 w0 L L 5L 4 = c1 x2 + c2 x3 + . x − x5 + x − x− 2 6 60EIL 2 2 2 To find c1 and c2 we compute ) # $3 L w0 y (x) = c1 + c2 x + 30Lx2 − 20x3 + 20 x − 60EIL 2 # ) # $2 L w0 60Lx − 60x2 + 60 x − y (x) = c2 + 60EIL 2 L x− 2 ′′ and # ′′′ L x− 2 $, $, . Then y ′′ (L) = y ′′′ (L) = 0 yields the system . w0 5w0 L2 5 c1 + c2 L + =0 30L3 − 20L3 + L3 = c1 + c2 L + 60EIL 2 24EI w0 w0 L c2 + [60L2 − 60L2 + 15L2 ] = c2 + = 0. 60EIL 4EI Solving for c1 and c2 we obtain c1 = w0 L2 /24EI and c2 = −w0 L/4EI. Thus ) # $5 w0 L 3 w0 L 5L 4 w0 L2 2 x − x + x − x5 + x − y(x) = 48EI 24EI 60EIL 2 2 # L x− 2 80. The differential equation is d4 y = w0 [1 − (x − L/2)]. dx4 Taking the Laplace transform of both sides and using y(0) = y ′ (0) = 0 we obtain 1 w0 1 0 s4 {y} − sy ′′ (0) − y ′′′ (0) = 1 − e−Ls/2 . EI s EI Letting y ′′ (0) = c1 and y ′′′ (0) = c2 we have {y} = so that 1 c2 w0 1 0 c1 −Ls/2 1 − e + + s3 s4 EI s5 ) # $4 1 L 1 1 w0 2 3 y(x) = c1 x + c2 x + x4 − x − 2 6 24 EI 2 # L x− 2 $, To find c1 and c2 we compute ) $2 # L w 1 0 2 ′′ x − x− y (x) = c1 + c2 x + 2 EI 2 220 # L x− 2 $, . . $, . 4.3 Translation Theorems Then y(L) = y ′′ (L) = 0 yields the system ) # $4 , L 1 1 w0 1 5w0 1 1 4 2 3 L − c1 L + c2 L + L4 = 0 = c1 L2 + c2 L3 + 2 6 24 EI 2 2 6 128EI ) # $2 , L 3w0 2 1 w0 = c1 + c2 L + c1 + c2 L + L2 − L = 0. 2 EI 2 8EI 9 57 Solving for c1 and c2 we obtain c1 = 128 w0 L2 /EI and c2 = − 128 w0 L/EI. Thus ) # $4 # $, w0 19 1 4 1 L L 9 2 2 3 y(x) = x− . L x − Lx + x − x− EI 256 256 24 24 2 2 81. (a) The temperature T of the cake inside the oven is modeled by where Tm dT = k(T − Tm ) dt is the ambient temperature of the oven. For 0 ≤ t ≤ 4, we have Tm = 70 + Hence for t ≥ 0, Tm = In terms of the unit step function, Tm = (70 + 57.5t)[1 − % 300 − 70 t = 70 + 57.5t. 4−0 70 + 57.5t, 300, (t − 4)] + 300 0≤t<4 t ≥ 4. (t − 4) = 70 + 57.5t + (230 − 57.5t) (t − 4). The initial-value problem is then dT = k[T − 70 − 57.5t − (230 − 57.5t) dt (b) Let t(s) = or (t − 4)], T (0) = 70. {T (t)}. Transforming the equation, using 230 − 57.5t = −57.5(t − 4) and Theorem 4.7, gives $ # 70 57.5 57.5 −4s st(s) − 70 = k t(s) − − 2 + 2 e s s s t(s) = 70k 57.5k 57.5k 70 − − 2 + 2 e−4s . s − k s(s − k) s (s − k) s (s − k) After using partial functions, the inverse transform is then # $ # $ 1 1 kt 1 1 k(t−4) T (t) = 70 + 57.5 − 57.5 +t− e +t−4− e k k k k (t − 4). Of course, the obvious question is: What is k? If the cake is supposed to bake for, say, 20 minutes, then T (20) = 300. That is, 300 = 70 + 57.5 # 1 1 + 20 − e20k k k $ − 57.5 # $ 1 1 16k . + 16 − e k k But this equation has no physically meaningful solution. This should be no surprise since the model predicts the asymptotic behavior T (t) → 300 as t increases. Using T (20) = 299 instead, we find, with the help of a CAS, that k ≈ −0.3. 82. In order to apply Theorem 4.7 we need the function to have the form f (t − a) rewrite the functions given in the forms shown below. 221 (t − a). To accomplish this 4.3 Translation Theorems (b) et = et−5+5 = e5 et−5 (a) 2t + 1 = 2(t − 1 + 1) + 1 = 2(t − 1) + 3 (d) t2 − 3t = (t − 2)2 + (t − 2) − 2 (c) cos t = − cos(t − π) {tekti } = 1/(s − ki)2 . Then, using Euler’s formula, 83. (a) From Theorem 4.6 we have {tekti } = = {t cos kt + it sin kt} = {t cos kt} + i {t sin kt} (s + ki)2 s2 − k 2 2ks 1 = 2 = 2 +i 2 . 2 2 2 (s − ki) (s + k ) (s + k 2 )2 (s + k 2 )2 Equating real and imaginary parts we have {t cos kt} = s2 − k 2 (s2 + k 2 )2 and {t sin kt} = (s2 2ks . + k 2 )2 (b) The Laplace transform of the differential equation is s2 Solving for {x} + ω 2 {x} = s2 s . + ω2 {x} = s/(s2 + ω 2 )2 . Thus x = (1/2ω)t sin ωt. {x} we obtain EXERCISES 4.4 Additional Operational Properties −10t d }=− ds 1. {te 3. {t cos 2t} = − 5. 6. 7. 8. # d ds 1 s + 10 # $ 1 = (s + 10)2 $ = s s2 + 4 s2 − 4 2 (s2 + 4) 2. d3 {t e } = (−1) ds3 4. {t sinh 3t} = − 3 t 3 d ds # # 1 s−1 $ = 3 s2 − 9 $ = 6 (s − 1)4 6s # $ 6s2 + 2 1 d2 = 3 2 2 ds s −1 (s2 − 1) ; < $ # # $ 2s s2 − 3 d2 s 1 − s2 d 2 {t cos t} = 2 = = 3 ds s2 + 1 ds (s2 + 1)2 (s2 + 1) # $ 9 2t : 6 12(s − 2) d = te sin 6t = − 2 ds (s − 2)2 + 36 [(s − 2)2 + 36] # $ 9 −3t : d s+3 (s + 3)2 − 9 te cos 3t = − = 2 2 ds (s + 3) + 9 [(s + 3)2 + 9] {t2 sinh t} = 9. The Laplace transform of the differential equation is s Solving for {y} + {y} = 2s . (s2 + 1)2 {y} we obtain {y} = 1 1 1 s 1 2s 1 1 s =− − + + + 2 . (s + 1)(s2 + 1)2 2 s + 1 2 s2 + 1 2 s2 + 1 (s2 + 1)2 (s + 1)2 222 2 (s2 − 9) 4.4 Thus 1 y(t) = − e−t − 2 1 = − e−t + 2 Additional Operational Properties 1 1 1 1 sin t + cos t + (sin t − t cos t) + t sin t 2 2 2 2 1 1 1 cos t − t cos t + t sin t. 2 2 2 10. The Laplace transform of the differential equation is s Solving for {y} − {y} = 2(s − 1) . ((s − 1)2 + 1)2 {y} we obtain {y} = Thus 2 . ((s − 1)2 + 1)2 y = et sin t − tet cos t. 11. The Laplace transform of the differential equation is s2 {y} − sy(0) − y ′ (0) + 9 Letting y(0) = 2 and y ′ (0) = 5 and solving for 3 {y} = s . s2 + 9 {y} = {y} we obtain 2 2s + 5s + 19s − 45 2s 5 s = 2 + 2 + 2 . 2 2 (s + 9) s + 9 s + 9 (s + 9)2 Thus 1 5 sin 3t + t sin 3t. 3 6 y = 2 cos 3t + 12. The Laplace transform of the differential equation is s2 Solving for {y} − sy(0) − y ′ (0) + {y} = 1 . s2 + 1 {y} we obtain {y} = s3 − s2 + s s 1 1 = 2 − 2 + 2 . 2 2 (s + 1) s + 1 s + 1 (s + 1)2 Thus y = cos t − sin t + # 1 1 sin t − t cos t 2 2 $ = cos t − 1 1 sin t − t cos t. 2 2 13. The Laplace transform of the differential equation is s2 {y} − sy(0) − y ′ (0) + 16 {y} = {cos 4t − cos 4t (t − π)} or by (16) of Section 4.3 in the text, (s2 + 16) s − e−πs + 16 s − e−πs =1+ 2 s + 16 {y} = 1 + s2 Thus {y} = and y= {cos 4(t + π)} {cos 4t} = 1 + s2 s s − 2 e−πs . + 16 s + 16 1 s s − 2 e−πs + s2 + 16 (s2 + 16)2 (s + 16)2 1 1 1 sin 4t + t sin 4t − (t − π) sin 4(t − π) 4 8 8 223 (t − π). 4.4 Additional Operational Properties 14. The Laplace transform of the differential equation is s2 {y} − sy(0) − y ′ (0) + (s2 + 1) or Thus {y} = 2 1− 0 t− 0 π1 π 13 + sin t t− 2 2 2 0 π 13 1 1 −πs/2 − e + e−πs/2 sin t + s s 2 1 1 −πs/2 + e−πs/2 {cos t} =s+ − e s s 1 1 s = s + − e−πs/2 + 2 e−πs/2 . s s s +1 {y} = s + s 1 1 s + − e−πs/2 + 2 e−πs/2 s2 + 1 s(s2 + 1) s(s2 + 1) (s + 1)2 # $ 1 1 s s s s + − − − e−πs/2 + 2 e−πs/2 = 2 s + 1 s s2 + 1 s s2 + 1 (s + 1)2 # $ 1 1 s s = − e−πs/2 − 2 e−πs/2 + 2 s s s +1 (s + 1)2 {y} = and 15. 0 > 0 π1 1 0 π1 π1 π 1? 0 t− + t− sin t − y = 1 − 1 − cos t − 2 2 2 2 2 0 0 π1 1 0 π1 π1 = 1 − (1 − sin t) t− − t− cos t t− . 2 2 2 2 16. y 1 0 t− π1 2 y 4 0.5 2 1 2 3 4 5 6 t 1 2 3 4 5 6 t -2 -0.5 -4 -1 17. From (7) of Section 4.2 in the text along with Theorem 4.8, d dY d {y ′′ } = − [s2 Y (s) − sy(0) − y ′ (0)] = −s2 − 2sY + y(0), ds ds ds so that the transform of the given second-order differential equation is the linear first-order differential equation in Y (s): 4 3 4 s2 Y ′ + 3sY = − 3 or Y′+ Y =− 5 . s s s The solution of the latter equation is Y (s) = 4/s4 + c/s3 , so {ty ′′ } = − y(t) = {Y (s)} = 2 3 c 2 t + t . 3 2 18. From Theorem 4.8 in the text d d dY {y ′ } = − [sY (s) − y(0)] = −s −Y ds ds ds so that the transform of the given second-order differential equation is the linear first-order differential equation {ty ′ } = − in Y (s): Y′+ # $ 10 3 − 2s Y = − . s s 224 4.4 Additional Operational Properties 2 Using the integrating factor s3 e−s , the last equation yields 5 c 2 + 3 es . s3 s But if Y (s) is the Laplace transform of a piecewise-continuous function of exponential order, we must have, in Y (s) = view of Theorem 4.5, lims→∞ Y (s) = 0. In order to obtain this condition we require c = 0. Hence % / 5 5 y(t) = = t2 . s3 2 19. 9 : 1 3! 6 = 5 1 ∗ t3 = 4 s s s 21. 9 : e−t ∗ et cos t = t 23. %! t 24. %! t 25. %! t 26. %! t 27. %! t 28. %! / = 1 s cos τ dτ / = τ e dτ 0 0 t−τ τe 0 29. 30. 31. 32. 33. 34. dτ 1 s / / = / = τ sin τ dτ 0 0 {et } = e−τ cos τ dτ 0 s−1 (s + 1) [(s − 1)2 + 1] = 1 s 1 s : t2 ∗ tet = 22. 9 : e2t ∗ sin t = 1 (s − 2)(s2 + 1) s 1 = 2 + 1) s +1 s(s2 : 1 s+1 s+1 e−t cos t = = s (s + 1)2 + 1 s (s2 + 2s + 2) # $ 2 1 d 1 −2s 1 = {t sin t} = − =− 2 s ds s2 + 1 s (s2 + 1)2 (s2 + 1) 9 {et } = {t} sin τ cos(t − τ ) dτ 9 1 s(s − 1) {cos t} = 2 s3 (s − 1)2 20. / = 1 s2 (s − 1) {sin t} {cos t} = s (s2 2 + 1) / / %! t # $ % ! t d 1 1 d 3s2 + 1 sin τ dτ = − sin τ dτ = − = t 2 2 ds ds s s + 1 s2 (s2 + 1) 0 0 / % ! t / %! t # $ d d 1 1 3s + 1 −τ −τ τ e dτ = − t τ e dτ = − = 2 2 ds ds s (s + 1) s (s + 1)3 0 0 % / % / ! t 1 1/(s − 1) eτ dτ = et − 1 = = s(s − 1) s 0 / % / ! t % 1/s(s − 1) 1 (eτ − 1)dτ = et − t − 1 = = s2 (s − 1) s 0 % / % / ! t 1 1 1/s2 (s − 1) (eτ − τ − 1)dτ = et − t2 − t − 1 = = 3 s (s − 1) s 2 0 / % 1 = teat , (8) in the text gives Using (s − a)2 % / ! t 1 1 τ eaτ dτ = 2 (ateat − eat + 1). = 2 s(s − a) a 0 35. (a) The result in (4) in the text is F (s) = {F (s)G(s)} = f ∗ g, so identify (s2 2k 3 + k 2 )2 and 225 G(s) = 4s . s2 + k 2 4.4 Additional Operational Properties Then f (t) = sin kt − kt cos kt so % 8k 3 s (s2 + k 2 )3 / = and g(t) = 4 cos kt {F (s)G(s)} = f ∗ g = 4 =4 ! t 0 ! t 0 f (τ )g(t − τ )dt (sin kτ − kτ cos kτ ) cos k(t − τ )dτ. Using a CAS to evaluate the integral we get % / 8k 3 s = t sin kt − kt2 cos kt. (s2 + k 2 )3 (b) Observe from part (a) that and from Theorem 4.8 that so 9 9 9 : t(sin kt − kt cos kt) = 8k 3 s , + k 2 )3 (s2 : tf (t) = −F ′ (s). We saw in (5) in the text that {sin kt − kt cos kt} = 2k 3 /(s2 + k 2 )2 , : d 8k 3 s 2k 3 t(sin kt − kt cos kt) = − = 2 . 2 2 2 ds (s + k ) (s + k 2 )3 36. The Laplace transform of the differential equation is s2 {y} + y 2s 1 . + {y} = 2 (s + 1) (s2 + 1)2 50 Thus 2s 1 + 2 (s2 + 1)2 (s + 1)3 and, using Problem 35 with k = 1, {y} = y= 5 -50 1 1 (sin t − t cos t) + (t sin t − t2 cos t). 2 4 37. The Laplace transform of the given equation is {f } + Solving for {f } we obtain {f } = s2 {t} {f } = {t}. 1 . Thus, f (t) = sin t. +1 38. The Laplace transform of the given equation is {f } = Solving for {f } we obtain {f } = {2t} − 4 {sin t} {f }. √ 2s2 + 2 8 2 1 5 √ + = . s2 (s2 + 5) 5 s2 5 5 s2 + 5 Thus f (t) = √ 8 2 t + √ sin 5 t. 5 5 5 39. The Laplace transform of the given equation is {f } = 9 : tet + 226 {t} {f }. 10 15 t 4.4 Solving for Additional Operational Properties {f } we obtain {f } = s2 1 1 3 1 1 1 1 2 = + + − . (s − 1)3 (s + 1) 8 s − 1 4 (s − 1)2 4 (s − 1)3 8 s+1 Thus 1 t 3 t 1 2 t 1 −t e + te + t e − e 8 4 4 8 f (t) = 40. The Laplace transform of the given equation is {f } + 2 Solving for {cos t} 9 {f } = 4 {f } we obtain {f } = : e−t + {sin t}. 4s2 + s + 5 4 7 2 = − +4 . (s + 1)3 s + 1 (s + 1)2 (s + 1)3 Thus f (t) = 4e−t − 7te−t + 4t2 e−t . 41. The Laplace transform of the given equation is {f } + Solving for {f } we obtain {f } = {1} {f } = {1}. 1 . Thus, f (t) = e−t . s+1 42. The Laplace transform of the given equation is {f } = Solving for {cos t} + {f } we obtain {f } = 9 e−t : {f }. 1 s + . s2 + 1 s2 + 1 Thus f (t) = cos t + sin t. 43. The Laplace transform of the given equation is {f } = = Solving for {1} + 1 8 1 + 2+ s s 3 {t} − {t3 } / % ! t 8 (t − τ )3 f (τ ) dτ 3 0 16 1 1 {f }. {f } = + 2 + 4 s s s {f } we obtain {f } = Thus s2 (s + 1) 1 1 3 1 1 2 1 s = + + + . 4 2 s − 16 8 s + 2 8 s − 2 4 s + 4 2 s2 + 4 f (t) = 1 1 −2t 3 2t 1 e + e + sin 2t + cos 2t. 8 8 4 2 44. The Laplace transform of the given equation is {t} − 2 Solving for {f } = {f } we obtain {f } = 9 et − e−t : {f }. s2 − 1 1 1 1 3! = − . 4 2 2s 2 s 12 s4 227 4.4 Additional Operational Properties Thus f (t) = 1 1 t − t3 . 2 12 45. The Laplace transform of the given equation is s Solving for {y} − y(0) = {1} − {sin t} − {1} {y}. {f } we obtain {y} = s2 − s + 1 1 2s 1 = 2 − . (s2 + 1)2 s + 1 2 (s2 + 1)2 Thus 1 t sin t. 2 y = sin t − 46. The Laplace transform of the given equation is s Solving for {y} − y(0) + 6 {f } we obtain {y} = {y} + 9 {1} {y} = 1 . Thus, y = te−3t . (s + 3)2 i 30 47. The differential equation is 0.1 di 1 + 3i + dt 0.05 di + 30i + 200 dt t ! 7 i(τ )dτ = 100 0 or 8 (t − 2) (t − 1) − 20 10 t ! 7 i(τ )dτ = 1000 0 8 (t − 2) , (t − 1) − where i(0) = 0. The Laplace transform of the differential equation is s Solving for {1}. {i} − y(0) + 30 200 {i} + s 0.5 1 1.5 2 2.5 3 t -10 -20 -30 1000 −s {i} = (e − e−2s ). s {i} we obtain {i} = 1000e−s − 1000e−2s = s2 + 30s + 200 # $ 100 100 − (e−s − e−2s ). s + 10 s + 20 Thus ; < i(t) = 100 e−10(t−1) − e−20(t−1) 48. The differential equation is 0.005 di 1 +i+ dt 0.02 ! di + 200i + 10,000 dt ! 0 (t − 2). i 2 t 0 or ; < (t − 1) − 100 e−10(t−2) − e−20(t−2) 7 i(τ )dτ = 100 t − (t − 1) 8 (t − 1) t 7 i(τ )dτ = 20,000 t − (t − 1) 8 (t − 1) , where i(0) = 0. The Laplace transform of the differential equation is # $ 1 1 −s 10,000 {i} = 20,000 2 − 2 e s {i} + 200 {i} + . s s s 228 1.5 1 0.5 0.5 1 1.5 2 t 4.4 Solving for Additional Operational Properties {i} we obtain . 20,000 2 2 200 −s {i} = (1 − e−s ). (1 − e ) = − − s(s + 100)2 s s + 100 (s + 100)2 Thus i(t) = 2 − 2e−100t − 200te−100t − 2 49. 50. {f (t)} = 1 1 − e−2as -! {f (t)} = 1 1 − e−2as ! a e−st dt − 0 a e−st dt = 0 (t − 1) + 2e−100(t−1) 2a ! a (t − 1) + 200(t − 1)e−100(t−1) (t − 1). . (1 − e−as )2 1 − e−as e−st dt = = −2as s(1 − e ) s(1 + e−as ) 1 s(1 + e−as ) 51. Using integration by parts, 1 1 − e−bs {f (t)} = 52. 53. {f (t)} = 1 1 − e−πs {f (t)} = 1 1 − e−2πs 54. 1 -! 1 {f (t)} = 1 − e−2s −st te dt + ! π e−st sin t dt = 0 ! 0 −st (2 − t)e a −st a te dt = b s . dt = # 1 1 − bs bs e − 1 $ . 1 − e−s s2 (1 − e−2s ) 1 1 eπs/2 + e−πs/2 πs = 2 · coth s2 + 1 eπs/2 − e−πs/2 s +1 2 π e−st sin t dt = 0 b 2 1 0 ! ! 1 1 · s2 + 1 1 − e−πs 55. The differential equation is L di/dt + Ri = E(t), where i(0) = 0. The Laplace transform of the equation is Ls From Problem 49 we have {i} + R {i} = {E(t)}. {E(t)} = (1 − e−s )/s(1 + e−s ). Thus (Ls + R) {i} = 1 − e−s s(1 + e−s ) and 1 1 1 − e−s 1 − e−s 1 = −s L s(s + R/L)(1 + e ) L s(s + R/L) 1 + e−s # $ 1 1 1 − (1 − e−s )(1 − e−s + e−2s − e−3s + e−4s − · · · ) = R s s + R/L # $ 1 1 1 = − (1 − 2e−s + 2e−2s − 2e−3s + 2e−4s − · · · ). R s s + R/L {i} = Therefore, 1 2 0 1 1 0 1 − e−Rt/L − 1 − e−R(t−1)/L (t − 1) R R 0 0 1 1 2 2 1 − e−R(t−2)/L 1 − e−R(t−3)/L + (t − 2) − R R ∞ 0 1 1 2 @ 1 0 1 − e−R(t−n)/L (t − n). = 1 − e−Rt/L + R R n=1 i(t) = 229 (t − 3) + · · · 4.4 Additional Operational Properties The graph of i(t) with L = 1 and R = 1 is shown below. i 1 0.5 1 2 3 4 t -0.5 -1 56. The differential equation is L di/dt + Ri = E(t), where i(0) = 0. The Laplace transform of the equation is Ls {i} + R {i} = {E(t)}. From Problem 51 we have {E(t)} = # $ 1 1 1 1 1 1 − s . = 2− s s e −1 s s es − 1 Thus (Ls + R) {i} = 1 1 1 − 2 s s es − 1 and 1 1 1 1 1 − L s2 (s + R/L) L s(s + R/L) es − 1 # $ # $ ; −s < 1 1 L 1 1 1 L 1 1 = e + e−2s + e−3s + · · · . − − + − 2 R s R s R s + R/L R s s + R/L {i} = Therefore $ L −Rt/L L − t− + e R R 1 1 0 − 1 − e−R(t−2)/L R $ # L L 1 t − + e−Rt/L − = R R R 1 i(t) = R # 1 1 0 (t − 1) 1 − e−R(t−1)/L R 1 1 0 1 − e−R(t−3)/L (t − 2) − (t − 3) − · · · R ∞ < 1 @; 1 − e−R(t−n)/L (t − n). R n=1 The graph of i(t) with L = 1 and R = 1 is shown below. i 1 0.5 1 2 3 4 t -0.5 -1 57. The differential equation is x′′ + 2x′ + 10x = 20f (t), where f (t) is the meander function in Problem 49 with 230 4.4 Additional Operational Properties a = π. Using the initial conditions x(0) = x′ (0) = 0 and taking the Laplace transform we obtain (s2 + 2s + 10) 20 1 (1 − e−πs ) s 1 + e−πs 20 (1 − e−πs )(1 − e−πs + e−2πs − e−3πs + · · ·) = s 20 (1 − 2e−πs + 2e−2πs − 2e−3πs + · · ·) = s ∞ 20 40 @ = (−1)n e−nπs . + s s n=1 {x(t)} = Then {x(t)} = ∞ @ 20 40 (−1)n e−nπs + s(s2 + 2s + 10) s(s2 + 2s + 10) n=1 . ∞ @ 2 2s + 4 4s + 8 n 4 = − 2 (−1) + − e−nπs s s + 2s + 10 n=1 s s2 + 2s + 10 . ∞ @ (s + 1) + 1 2 2(s + 1) + 2 n 1 (−1) +4 − e−nπs = − 2+9 s (s + 1)2 + 9 s (s + 1) n=1 and $ # ∞ @ 1 x(t) = 2 1 − e−t cos 3t − e−t sin 3t + 4 (−1)n 1 − e−(t−nπ) cos 3(t − nπ) 3 n=1 . 1 − e−(t−nπ) sin 3(t − nπ) (t − nπ). 3 The graph of x(t) on the interval [0, 2π) is shown below. x 3 π 2π t −3 58. The differential equation is x′′ + 2x′ + x = 5f (t), where f (t) is the square wave function with a = π. Using the initial conditions x(0) = x′ (0) = 0 and taking the Laplace transform, we obtain (s2 + 2s + 1) {x(t)} = = 5 5 1 = (1 − e−πs + e−2πs − e−3πs + e−4πs − · · ·) −πs s 1+e s ∞ 5@ (−1)n e−nπs . s n=0 Then {x(t)} = # $ ∞ ∞ @ @ 5 1 1 1 n n −nπs (−1) (−1) e = 5 − − e−nπs 2 s(s + 1)2 n=0 s s + 1 (s + 1) n=0 231 4.4 Additional Operational Properties and x(t) = 5 ∞ @ n=0 (−1)n (1 − e−(t−nπ) − (t − nπ)e−(t−nπ) ) (t − nπ). The graph of x(t) on the interval [0, 4π) is shown below. x 5 2π 4π t −5 59. f (t) = − 1 t % / d 1 [ln(s − 3) − ln(s + 1)] = − ds t % 1 1 − s−3 s+1 / =− 60. The transform of Bessel’s equation is − < 1 ; 3t e − e−t t d 2 d [s Y (s) − sy(0) − y ′ (0)] + sY (s) − y(0) − Y (s) = 0 ds ds or, after simplifying and using the initial condition, (s2 + 1)Y ′ + sY = 0. This equation is both separable and √ linear. Solving gives Y (s) = c/ s2 + 1 . Now Y (s) = {J0 (t)}, where J0 has a derivative that is continuous and of exponential order, implies by Problem 46 of Exercises 4.2 that 1 = J0 (0) = lim sY (s) = c lim √ s→∞ s→∞ s2 s =c + k2 so c = 1 and Y (s) = √ 1 s2 + 1 or {J0 (t)} = √ 1 . s2 + 1 61. (a) Using Theorem 4.8, the Laplace transform of the differential equation is − d 2 d [s Y − sy(0) − y ′ (0)] + sY − y(0) + [sY − y(0)] + nY ds ds d d = − [s2 Y ] + sY + [sY ] + nY ds ds $ # $ # dY dY − 2sY + sY + s + Y + nY = −s2 ds ds # $ dY 2 = (s − s ) + (1 + n − s)Y = 0. ds Separating variables, we find dY 1+n−s = ds = Y s2 − s # n 1+n − s−1 s ln Y = n ln(s − 1) − (1 + n) ln s + c Y = c1 (s − 1)n . s1+n 232 $ ds 4.4 Additional Operational Properties Since the differential equation is homogeneous, any constant multiple of a solution will still be a solution, so for convenience we take c1 = 1. The following polynomials are solutions of Laguerre’s differential equation: % / 1 n = 0 : L0 (t) = =1 s % / % / s−1 1 1 n = 1 : L1 (t) = = =1−t − s2 s s2 / % / % 2 1 1 1 (s − 1)2 n = 2 : L2 (t) = − 2 + 3 = 1 − 2t + t2 = 3 s s s s 2 % / % / 3 (s − 1) 3 3 1 1 1 3 n = 3 : L3 (t) = − + 3 − 4 = 1 − 3t + t2 − t3 = s4 s s2 s s 2 6 / % / % 4 6 4 1 1 (s − 1)4 − + 3− 4+ 5 = n = 4 : L4 (t) = s5 s s2 s s s 2 1 = 1 − 4t + 3t2 − t3 + t4 . 3 24 (b) Letting f (t) = tn e−t we note that f (k) (0) = 0 for k = 0, 1, 2, . . . , n − 1 and f (n) (0) = n!. Now, by the first translation theorem, % t n / " e d n −t 1 1 t e {et f (n) (t)} = {f (n) (t)} "s→s−1 = n n! dt n! n! ? 1> n s {tn e−t } − sn−1 f (0) − sn−2 f ′ (0) − · · · − f (n−1) (0) = n! s→s−1 ? 1> n n −t = s {t e } n! s→s−1 . (s − 1)n 1 n n! = = s = Y, n+1 n! (s + 1) sn+1 s→s−1 where Y = {Ln (t)}. Thus Ln (t) = et dn n −t (t e ), n! dtn n = 0, 1, 2, . . . . 62. The output for the first three lines of the program are 9y[t] + 6y ′ [t] + y ′′ [t] == t sin[t] 2s (1 + s2 )2 # $ −11 − 4s − 22s2 − 4s3 − 11s4 − 2s5 Y →− (1 + s2 )2 (9 + 6s + s2 ) 1 − 2s + 9Y + s2 Y + 6(−2 + sY ) == The fourth line is the same as the third line with Y → removed. The final line of output shows a solution involving complex coefficients of eit and e−it . To get the solution in more standard form write the last line as two lines: euler={Eˆ(It)−>Cos[t] + I Sin[t], Eˆ(-It)−>Cos[t] - I Sin[t]} InverseLaplaceTransform[Y, s, t]/.euler//Expand We see that the solution is # $ 487 247 1 y(t) = + t e−3t + (13 cos t − 15t cos t − 9 sin t + 20t sin t) . 250 50 250 63. The solution is √ 1 1 y(t) = et − e−t/2 cos 15 t − 6 6 233 = √ 3/5 −t/2 e sin 15 t. 6 4.4 Additional Operational Properties 64. The solution is q(t) = 1 − cos t + (6 − 6 cos t) (t − 3π) − (4 + 4 cos t) (t − π). q 5 Π -5 3Π t 5Π EXERCISES 4.5 The Dirac Delta Function 1. The Laplace transform of the differential equation yields {y} = so that 1 −2s e s−3 y = e3(t−2) (t − 2). 2. The Laplace transform of the differential equation yields {y} = e−s 2 + s+1 s+1 so that y = 2e−t + e−(t−1) (t − 1). 3. The Laplace transform of the differential equation yields < 1 ; {y} = 2 1 + e−2πs s +1 so that y = sin t + sin t (t − 2π). 4. The Laplace transform of the differential equation yields 1 4 {y} = e−2πs 2 4 s + 16 so that 1 1 y = sin 4(t − 2π) (t − 2π) = sin 4t 4 4 (t − 2π). 5. The Laplace transform of the differential equation yields 1 1 0 −πs/2 {y} = 2 e + e−3πs/2 s +1 so that # $ # $ 0 π1 0 π1 3π 3π y = sin t − t− + sin t − t− 2 2 2 2 0 0 π1 π1 = − cos t t− + cos t t− . 2 2 234 4.5 The Dirac Delta Function 6. The Laplace transform of the differential equation yields {y} = s 1 + (e−2πs + e−4πs ) s2 + 1 s2 + 1 so that y = cos t + sin t[ (t − 2π) + (t − 4π)]. 7. The Laplace transform of the differential equation yields . 1 1 1 1 1 −s {y} = 2 (1 + e ) = − (1 + e−s ) s + 2s 2 s 2 s+2 so that . 1 1 −2t 1 1 −2(t−1) y= − e + − e (t − 1). 2 2 2 2 8. The Laplace transform of the differential equation yields {y} = . s+1 1 1 1 3 1 3 1 1 1 1 1 −2s −2s + e + e = − − − s2 (s − 2) s(s − 2) 4 s − 2 4 s 2 s2 2 s−2 2 s so that y= . 3 2t 3 1 1 1 e − − t + e2(t−2) − 4 4 2 2 2 (t − 2). 9. The Laplace transform of the differential equation yields {y} = 1 e−2πs (s + 2)2 + 1 so that y = e−2(t−2π) sin t (t − 2π). 10. The Laplace transform of the differential equation yields {y} = 1 e−s (s + 1)2 so that y = (t − 1)e−(t−1) (t − 1). 11. The Laplace transform of the differential equation yields {y} = = so that s2 e−πs + e−3πs 4+s + 2 + 4s + 13 s + 4s + 13 ; −πs < 2 s+2 1 3 3 + + e + e−3πs 2 2 2 2 2 2 3 (s + 2) + 3 (s + 2) + 3 3 (s + 2) + 3 y= 1 2 −2t e sin 3t + e−2t cos 3t + e−2(t−π) sin 3(t − π) 3 3 1 + e−2(t−3π) sin 3(t − 3π) (t − 3π). 3 (t − π) 12. The Laplace transform of the differential equation yields {y} = 1 e−2s + e−4s + 2 (s − 1) (s − 6) (s − 1)(s − 6) =− . < ; −2s 1 1 1 1 1 1 1 1 1 1 e + e−4s − + + − + 2 25 s − 1 5 (s − 1) 25 s − 6 5 s−1 5 s−6 235 4.5 The Dirac Delta Function so that . . 1 t 1 t 1 6t 1 t−2 1 6(t−2) 1 t−4 1 6(t−4) y = − e − te + e + − e + e + e (t − 2) + − e (t − 4). 25 5 25 5 5 5 5 13. The Laplace transform of the differential equation yields {y} = 1 2 ′′ 1 3! ′′′ 1 P0 3! −Ls/2 y (0) + y (0) + e 3 4 2 s 6 s 6 EI s4 so that y= 1 1 P0 1 ′′ y (0)x2 + y ′′′ (0)x3 + 2 6 6 EI Using y ′′ (L) = 0 and y ′′′ (L) = 0 we obtain # x− L 2 # $3 L 1 P0 L 2 1 P0 3 1 P0 y= x − x + x− 4 EI 6 EI 6 EI 2 $ # ⎧ L P0 L 2 1 3 ⎪ ⎪ ⎪ ⎨ EI 4 x − 6 x , 0 ≤ x < 2 = # $ 2 ⎪ 1 L L ⎪ ⎪ P0 L ⎩ x− , ≤ x ≤ L. 4EI 2 12 2 $3 # # x− L x− 2 L 2 $ . $ . $ 14. From Problem 13 we know that 1 1 1 P0 y = y ′′ (0)x2 + y ′′′ (0)x3 + 2 6 6 EI # L x− 2 $3 # L x− 2 Using y(L) = 0 and y ′ (L) = 0 we obtain 1 P0 L 2 1 y= x − 16 EI 12 # ⎧ P0 L 2 ⎪ ⎪ ⎪ ⎨ EI 16 x − = # ⎪ ⎪ ⎪ P0 L x2 − ⎩ EI 16 # $3 # $ L L P0 3 1 P0 x− x− x + EI 6 EI 2 2 $ L 1 3 0≤x< x , 12 2 # $3 $ L 1 3 1 P0 L , x− x + ≤ x ≤ L. 12 6 EI 2 2 15. You should disagree. Although formal manipulations of the Laplace transform lead to y(t) = 31 e−t sin 3t in both cases, this function does not satisfy the initial condition y ′ (0) = 0 of the second initial-value problem. 236 4.6 Systems of Linear Differential Equations EXERCISES 4.6 Systems of Linear Differential Equations 1. Taking the Laplace transform of the system gives s s {x} = − {x} + {y} − 1 = 2 {y} {x} so that {x} = 1 1 1 1 1 = − (s − 1)(s + 2) 3 s−1 3 s+2 {y} = 2 2 1 1 1 1 + = + . s s(s − 1)(s + 2) 3 s−1 3 s+2 and Then x= 1 t 1 −2t e − e 3 3 and y= 2 t 1 −2t e + e . 3 3 2. Taking the Laplace transform of the system gives s {x} − 1 = 2 s {y} − 1 = 8 1 s−1 1 {x} − 2 s {y} + so that {y} = and y= s3 + 7s2 − s + 1 1 1 8 173 1 53 1 1 = − + − 2 s(s − 1)(s − 16) 16 s 15 s − 1 96 s − 4 160 s + 4 1 8 173 4t 53 −4t − et + e − e . 16 15 96 160 Then x= 1 1 173 4t 53 −4t 1 ′ 1 y + t = t − et + e + e . 8 8 8 15 192 320 3. Taking the Laplace transform of the system gives s {x} + 1 = s {y} − 2 = 5 {x} − 2 {y} {x} − {y} so that {x} = −s − 5 s 5 3 =− 2 − 2 s +9 s + 9 3 s2 + 9 and x = − cos 3t − 5 sin 3t. 3 Then y= 1 1 7 x − x′ = 2 cos 3t − sin 3t. 2 2 3 237 4.6 Systems of Linear Differential Equations 4. Taking the Laplace transform of the system gives {x} + s {y} = 1 s {x} + (s − 1) {y} = 1 s−1 (s + 3) (s − 1) so that {y} = 5s − 1 1 1 1 1 4 1 =− + + 3s(s − 1)2 3 s 3 s − 1 3 (s − 1)2 {x} = 1 − 2s 1 1 1 1 1 1 = − − . 3s(s − 1)2 3 s 3 s − 1 3 (s − 1)2 and Then x= 1 1 t 1 t − e − te 3 3 3 1 1 4 y = − + et + tet . 3 3 3 and 5. Taking the Laplace transform of the system gives (2s − 2) (s − 3) {x} + s {x} + (s − 3) 1 s 2 {y} = s {y} = so that {x} = −s − 3 1 1 5 1 2 =− + − s(s − 2)(s − 3) 2 s 2 s−2 s−3 {y} = 3s − 1 1 1 5 1 8 1 =− − + . s(s − 2)(s − 3) 6 s 2 s−2 3 s−3 and Then 1 5 x = − + e2t − 2e3t 2 2 1 5 8 y = − − e2t + e3t . 6 2 3 and 6. Taking the Laplace transform of the system gives (s + 1) {x} − (s − 1) {y} = −1 s {x} + (s + 2) {y} = 1 so that s + 1/2 s + 1/2 √ = s2 + s + 1 (s + 1/2)2 + ( 3/2)2 √ √ −3/2 3/2 √ {x} = 2 =− 3 . 2 s +s+1 (s + 1/2) + ( 3/2)2 {y} = and Then −t/2 y=e √ 3 t cos 2 and √ x = − 3 e−t/2 sin 7. Taking the Laplace transform of the system gives (s2 + 1) − {x} − {x} + (s2 + 1) {y} = −2 {y} = 1 so that {x} = and −2s2 − 1 1 1 3 1 =− 2 − s4 + 2s2 2 s 2 s2 + 2 238 √ 3 t. 2 4.6 Systems of Linear Differential Equations √ 1 3 x = − t − √ sin 2 t. 2 2 2 Then √ 3 1 y = x′′ + x = − t + √ sin 2 t. 2 2 2 8. Taking the Laplace transform of the system gives (s + 1) 4 {x} + {y} = 1 {x} − (s + 1) {y} = 1 so that {x} = s+2 s+1 1 2 = + s2 + 2s + 5 (s + 1)2 + 22 2 (s + 1)2 + 22 and {y} = Then s2 s+1 2 −s + 3 =− +2 . 2 2 + 2s + 5 (s + 1) + 2 (s + 1)2 + 22 1 x = e−t cos 2t + e−t sin 2t 2 and y = −e−t cos 2t + 2e−t sin 2t. 9. Adding the equations and then subtracting them gives 1 d2 x = t2 + 2t 2 dt 2 d2 y 1 = t2 − 2t. dt2 2 Taking the Laplace transform of the system gives 1 1 3! 1 4! {x} = 8 + + 5 s 24 s 3 s4 and {y} = so that x=8+ 1 4 1 3 t + t 24 3 1 4! 1 3! − 5 24 s 3 s4 and y= 1 4 1 3 t − t . 24 3 10. Taking the Laplace transform of the system gives (s − 4) (s + 2) {x} + s3 {x} − 2s3 {y} = s2 6 +1 {y} = 0 so that {x} = 4 4 1 4 s 8 1 = − − (s − 2)(s2 + 1) 5 s − 2 5 s2 + 1 5 s2 + 1 {y} = 2s + 4 2 1 1 1 2 6 s 8 1 = − 2 −2 3 + − + . s3 (s − 2)(s2 + 1) s s s 5 s − 2 5 s2 + 1 5 s2 + 1 and Then x= and 8 4 2t 4 e − cos t − sin t 5 5 5 1 6 8 y = 1 − 2t − 2t2 + e2t − cos t + sin t. 5 5 5 239 4.6 Systems of Linear Differential Equations 11. Taking the Laplace transform of the system gives s2 {x} + 3(s + 1) s2 so that {x} = − {x} + 3 {y} = 2 {y} = 1 (s + 1)2 1 1 1 2 1 2s + 1 = + 2+ − . 3 + 1) s s 2 s s+1 s3 (s Then 1 x = 1 + t + t2 − e−t 2 and y= 1 −t 1 ′′ 1 1 1 te − x = te−t + e−t − . 3 3 3 3 3 12. Taking the Laplace transform of the system gives {x} + 2 {y} = 2e−s s {x} + (s + 1) {y} = 1 e−s + 2 s (s − 4) −3 so that 1 −1/2 + e−s (s − 1)(s − 2) (s − 1)(s − 2) . 1 1 1 1 1 1 −s − − +e + = 2 s−1 2 s−2 s−1 s−2 {x} = and e−s s/4 − 1 −s/2 + 2 + + e−s s (s − 1)(s − 2) (s − 1)(s − 2) . 1 1 3 1 1 3 1 −s 1 − +e − + . = 4 s−1 2 s−2 s 2 s−1 s−2 {y} = Then ? 1 t 1 2t > t−1 + e2(t−1) (t − 1) e − e + −e 2 2 . 3 1 3 y = et − e2t + 1 − et−1 + e2(t−1) (t − 1). 4 2 2 x= and 13. The system is x′′1 = −3x1 + 2(x2 − x1 ) x′′2 = −2(x2 − x1 ) x1 (0) = 0 x′1 (0) = 1 x2 (0) = 1 x′2 (0) = 0. Taking the Laplace transform of the system gives (s2 + 5) −2 {x1 } − 2 {x1 } + (s2 + 2) 240 {x2 } = 1 {x2 } = s 4.6 so that and Systems of Linear Differential Equations √ 2 s 1 1 2 s 4 6 s2 + 2s + 2 {x1 } = 4 = + − + √ 2 2 2 2 2 s + 7s + 6 5 s +1 5 s +1 5 s +6 5 6 s +6 √ 6 s3 + 5s + 2 4 s 2 1 1 s 2 {x2 } = 2 = + + − √ 2 . 2 2 2 2 (s + 1)(s + 6) 5 s +1 5 s +1 5 s +6 5 6 s +6 Then x1 = √ √ 1 2 2 4 cos t + sin t − cos 6 t + √ sin 6 t 5 5 5 5 6 x2 = √ √ 2 2 1 4 cos t + sin t + cos 6 t − √ sin 6 t. 5 5 5 5 6 and 14. In this system x1 and x2 represent displacements of masses m1 and m2 from their equilibrium positions. Since the net forces acting on m1 and m2 are −k1 x1 + k2 (x2 − x1 ) and − k2 (x2 − x1 ) − k3 x2 , respectively, Newton’s second law of motion gives m1 x′′1 = −k1 x1 + k2 (x2 − x1 ) m2 x′′2 = −k2 (x2 − x1 ) − k3 x2 . Using k1 = k2 = k3 = 1, m1 = m2 = 1, x1 (0) = 0, x1 (0) = −1, x2 (0) = 0, and x′2 (0) = 1, and taking the Laplace transform of the system, we obtain (2 + s2 ) {x1 } − {x1 } − (2 + s2 ) so that 1 +3 and √ 1 x1 = − √ sin 3 t 3 and {x1 } = − Then s2 {x2 } = −1 {x2 } = −1 {x2 } = s2 1 . +3 √ 1 x2 = √ sin 3 t. 3 15. (a) By Kirchhoff’s first law we have i1 = i2 + i3 . By Kirchhoff’s second law, on each loop we have E(t) = Ri1 + L1 i′2 and E(t) = Ri1 + L2 i′3 or L1 i′2 + Ri2 + Ri3 = E(t) and L2 i′3 + Ri2 + Ri3 = E(t). (b) Taking the Laplace transform of the system 0.01i′2 + 5i2 + 5i3 = 100 0.0125i′3 + 5i2 + 5i3 = 100 gives {i2 } + 500 {i3 } = 10,000 s {i2 } + (s + 400) {i3 } = 8,000 s (s + 500) 400 so that {i3 } = Then i3 = 80 80 −900t − e 9 9 s2 and 80 1 80 1 8,000 = − . + 900s 9 s 9 s + 900 i2 = 20 − 0.0025i′3 − i3 = 241 100 100 −900t − e . 9 9 4.6 Systems of Linear Differential Equations (c) i1 = i2 + i3 = 20 − 20e−900t 16. (a) Taking the Laplace transform of the system i′2 + i′3 + 10i2 = 120 − 120 −10i′2 + 5i′3 (t − 2) + 5i3 = 0 gives (s + 10) −10s {i2 } + s {i2 } + 5(s + 1) {i3 } = < 120 ; 1 − e−2s s {i3 } = 0 so that {i2 } = and {i3 } = Then and . ; < < 48 60 12 ; 120(s + 1) −2s 1 − e = − + 1 − e−2s 2 (3s + 11s + 10)s s + 5/3 s + 2 s . < < ; 240 240 ; 240 −2s 1 − e−2s . − 1 − e = 2 3s + 11s + 10 s + 5/3 s + 2 > ? i2 = 12 + 48e−5t/3 − 60e−2t − 12 + 48e−5(t−2)/3 − 60e−2(t−2) (t − 2) > ? i3 = 240e−5t/3 − 240e−2t − 240e−5(t−2)/3 − 240e−2(t−2) (t − 2). > ? (b) i1 = i2 + i3 = 12 + 288e−5t/3 − 300e−2t − 12 + 288e−5(t−2)/3 − 300e−2(t−2) (t − 2) 17. Taking the Laplace transform of the system i′2 + 11i2 + 6i3 = 50 sin t i′3 + 6i2 + 6i3 = 50 sin t gives (s + 11) 6 {i2 } + 6 {i2 } + (s + 6) 50 +1 50 {i3 } = 2 s +1 {i3 } = s2 so that {i2 } = 1 1 20 1 375 145 s 85 50s =− + + + . 2 2 2 (s + 2)(s + 15)(s + 1) 13 s + 2 1469 s + 15 113 s + 1 113 s + 1 Then and i2 = − i3 = 20 −2t 375 −15t 145 85 e + e + cos t + sin t 13 1469 113 113 25 11 30 −2t 250 −15t 280 1 810 sin t − i′2 − i2 = e + e − cos t + sin t. 3 6 6 13 1469 113 113 18. Taking the Laplace transform of the system 0.5i′1 + 50i2 = 60 0.005i′2 + i2 − i1 = 0 242 4.6 Systems of Linear Differential Equations gives s {i2 } = {i2 } = {i1 } + (s + 200) −200 so that 120 s {i2 } = 0 {i1 } + 100 24,000 6 1 6 6 s + 100 100 − . = − s(s2 + 200s + 20,000) 5 s 5 (s + 100)2 + 1002 5 (s + 100)2 + 1002 Then i2 = and 6 6 6 −100t − e cos 100t − e−100t sin 100t 5 5 5 i1 = 0.005i′2 + i2 = 6 6 −100t cos 100t. − e 5 5 19. Taking the Laplace transform of the system 2i′1 + 50i2 = 60 0.005i′2 + i2 − i1 = 0 gives 2s {i1 } + 50 {i1 } + (s + 200) −200 60 s {i2 } = 0 {i2 } = so that {i2 } = 6,000 s(s2 + 200s + 5,000) √ √ 6 2 s + 100 50 2 6 1 6 √ √ − − . = 5 s 5 (s + 100)2 − (50 2 )2 5 (s + 100)2 − (50 2 )2 Then √ √ √ 6 2 −100t 6 6 −100t i2 = − e cosh 50 2 t − e sinh 50 2 t 5 5 5 √ √ √ 6 6 −100t 9 2 −100t ′ i1 = 0.005i2 + i2 = − e cosh 50 2 t − e sinh 50 2 t. 5 5 10 and 20. (a) Using Kirchhoff’s first law we write i1 = i2 + i3 . Since i2 = dq/dt we have i1 − i3 = dq/dt. Using Kirchhoff’s second law and summing the voltage drops across the shorter loop gives E(t) = iR1 + so that i1 = 1 q, C 1 1 E(t) − q. R1 R1 C Then dq 1 1 = i1 − i3 = E(t) − q − i3 dt R1 R1 C and R1 1 dq + q + R1 i3 = E(t). dt C 243 (1) 4.6 Systems of Linear Differential Equations Summing the voltage drops across the longer loop gives E(t) = i1 R1 + L di3 + R2 i3 . dt Combining this with (1) we obtain i1 R1 + L or L 1 di3 + R2 i3 = i1 R1 + q dt C di3 1 + R2 i3 − q = 0. dt C (b) Using L = R1 = R2 = C = 1, E(t) = 50e−t (t − 1) = 50e−1 e−(t−1) the Laplace transform of the system we obtain {i3 } = (t − 1), q(0) = i3 (0) = 0, and taking 50e−1 −s e s+1 (s + 1) {q} + (s + 1) {i3 } − {q} = 0, {q} = 50e−1 e−s (s + 1)2 + 1 so that and q(t) = 50e−1 e−(t−1) sin(t − 1) (t − 1) = 50e−t sin(t − 1) (t − 1). 21. (a) Taking the Laplace transform of the system 4θ1′′ + θ2′′ + 8θ1 = 0 θ1′′ + θ2′′ + 2θ2 = 0 gives ; < 4 s2 + 2 s2 so that ; or {θ1 } + s2 ; < {θ1 } + s2 + 2 <; < 3s2 + 4 s2 + 4 {θ2 } = Then θ2 = θ1 = (b) θ1 {θ2 } = 0 {θ2 } = −3s3 1 3 s s − . 2 2 s + 4/3 2 s2 + 4 1 2 3 cos √ t − cos 2t 2 2 3 so that {θ2 } = 3s θ1′′ = −θ2′′ − 2θ2 and 2 3 1 cos √ t + cos 2t. 4 4 3 θ2 2 2 1 1 −1 3π 6π t −1 −2 −2 244 3π 6π t 4.6 Systems of Linear Differential Equations Mass m2 has extreme displacements of greater magnitude. Mass m1 first passes through its equilibrium position at about t = 0.87, and mass m2 first passes through its equilibrium position at about t = 0.66. √ √ The motion of the pendulums is not periodic since cos(2t/ 3 ) has period 3 π, cos 2t has period π, and √ the ratio of these periods is 3 , which is not a rational number. θ2 (c) The Lissajous curve is plotted for 0 ≤ t ≤ 30. 2 1 -1-0.5 0.5 1 θ1 -1 -2 (d) t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t θ1 θ2 1 2 3 4 5 6 7 8 9 10 -0.2111 -0.6585 0.4830 -0.1325 -0.4111 0.8327 0.0458 -0.9639 0.3534 0.4370 0.8263 0.6438 -1.9145 0.1715 1.6951 -0.8662 -0.3186 0.9452 -1.2741 -0.3502 t=9 (e) Using a CAS to solve θ1 (t) = θ2 (t) we see that θ1 = θ2 (so that the double pendulum is straight out) when t is about 0.75 seconds. t=10 t=0.75 (f ) To make a movie of the pendulum it is necessary to locate the mass in the plane as a function of time. Suppose that the upper arm is attached to the origin and that the equilibrium position lies along the 245 4.6 Systems of Linear Differential Equations negative y-axis. Then mass m1 is at (x, (t), y1 (t)) and mass m2 is at (x2 (t), y2 (t)), where x1 (t) = 16 sin θ1 (t) and y1 (t) = −16 cos θ1 (t) x2 (t) = x1 (t) + 16 sin θ2 (t) and y2 (t) = y1 (t) − 16 cos θ2 (t). and A reasonable movie can be constructed by letting t range from 0 to 10 in increments of 0.1 seconds. CHAPTER 4 REVIEW EXERCISES 1. 1 {f (t)} = ! 4 {f (t)} = ! −st te dt + ∞ 1 0 2. ! e−st dt = 2 (2 − t)e−st dt = 1 2 − 2 e−s s2 s < 1 ; −2s e − e−4s s 3. False; consider f (t) = t−1/2 . 4. False, since f (t) = (et )10 = e10t . 5. True, since lims→∞ F (s) = 1 ̸= 0. (See Theorem 4.5 in the text.) 6. False; consider f (t) = 1 and g(t) = 1. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 9 9 : e−7t = 1 s+7 : te−7t = 1 (s + 7)2 2 s2 + 4 9 −3t : 2 e sin 2t = (s + 3)2 + 4 . d 4s 2 {t sin 2t} = − = 2 ds s2 + 4 (s + 4)2 {sin 2t} = {sin 2t % (t − π)} = (t − π)} = s2 2 e−πs +4 / 1 5! 1 = t5 6 s6 6 % / % / 1 1 1 1 = = et/3 3s − 1 3 s − 1/3 3 / / % % 2 1 1 1 = = t2 e5t (s − 5)3 2 (s − 5)3 2 / % / % √ 1 1 √ 1 1 1 1 1 √ √ √ √ = − √ e− 5 t + √ e 5 t + = − 2 s −5 2 5 s+ 5 2 5 s− 5 2 5 2 5 20 s6 / {sin 2(t − π) = % 246 CHAPTER 4 REVIEW EXERCISES % / s = s2 − 10s + 29 % / 1 −5s e = (t − 5) s2 / % s + π −s = e s2 + π 2 17. 18. 19. 20. 21. 22. 23. 24. % s−5 5 2 + 2 2 (s − 5) + 2 2 (s − 5)2 + 22 5 = e5t cos 2t + e5t sin 2t 2 (t − 5) % s π e−s + 2 e−s s2 + π 2 s + π2 = cos π(t − 1) / 1 L 1 = 2 L2 s2 + n2 π 2 L nπ 9 −5t : e exists for s > −5. % 9 / / (t − 1) + sin π(t − 1) (t − 1) % / 1 nπ/L nπ = sin t s2 + (n2 π 2 )/L2 Lnπ L : d te8t f (t) = − F (s − 8). ds {eat f (t − k) (t − k)} = e−ks {ea(t+k) f (t)} = e−ks eak {eat f (t)} = e−k(s−a) F (s − a) / %! t 1 F (s − a) eaτ f (τ ) dτ = {eat f (t)} = , whereas s s 0 " /" / %! t % ! t " F (s − a) F (s) "" at " = . f (τ ) dτ " f (τ ) dτ = e = " s s−a 0 0 s→s−a s→s−a 25. f (t) (t − t0 ) 26. f (t) − f (t) 27. f (t − t0 ) (t − t0 ) (t − t0 ) 28. f (t) − f (t) (t − t0 ) + f (t) 29. f (t) = t − [(t − 1) + 1] (t − t1 ) (t − 1) + (t − 1) − (t − 4) = t − (t − 1) (t − 1) − 1 1 1 − 2 e−s − e−4s 2 s s s 9 t : 1 1 −4(s−1) 1 − e−(s−1) − e e f (t) = 2 2 (s − 1) (s − 1) s−1 (t − 4) {f (t)} = 30. f (t) = sin t (t − π) − sin t (t − 3π) = − sin(t − π) (t − π) + sin(t − 3π) 1 1 {f (t)} = − 2 e−πs + 2 e−3πs s +1 s +1 9 t : 1 1 e−π(s−1) + e−3π(s−1) e f (t) = − (s − 1)2 + 1 (s − 1)2 + 1 31. f (t) = 2 − 2 (t − 2) + [(t − 2) + 2] (t − 2) = 2 + (t − 2) 1 2 {f (t)} = + 2 e−2s s s 9 t : 2 1 e f (t) = + e−2(s−1) s − 1 (s − 1)2 32. f (t) = t − t (t − 2) (t − 1) + (2 − t) (t − 1) − (2 − t) (t − 2) = t − 2(t − 1) 1 2 1 {f (t)} = 2 − 2 e−s + 2 e−2s s s s 9 t : 1 2 1 e f (t) = − e−(s−1) + e−2(s−1) (s − 1)2 (s − 1)2 (s − 1)2 247 (t − 3π) (t − 1) + (t − 2) (t − 2) CHAPTER 4 REVIEW EXERCISES 33. Taking the Laplace transform of the differential equation we obtain {y} = so that 5 2 1 + (s − 1)2 2 (s − 1)3 1 y = 5tet + t2 et . 2 34. Taking the Laplace transform of the differential equation we obtain {y} = = (s − 1 − 8s + 20) 1)2 (s2 6 5 1 1 1 s−4 2 6 − + + 2 2 2 169 s − 1 13 (s − 1) 169 (s − 4) + 2 338 (s − 4)2 + 22 so that y= 6 t 1 6 4t 5 4t e + tet − e cos 2t + e sin 2t. 169 13 169 338 35. Taking the Laplace transform of the given differential equation we obtain s3 + 6s2 + 1 1 2 − e−2s − e−2s + 1)(s + 5) s2 (s + 1)(s + 5) s(s + 1)(s + 5) 3 1 13 1 6 1 1 1 − · =− · + · 2 + · 25 s 5 s 2 s + 1 50 s + 5 $ # 1 1 1 1 6 1 1 1 e−2s − · − − · + · 2+ · 25 s 5 s 4 s + 1 100 s + 5 # $ 2 1 1 1 1 1 − · − · + · e−2s 5 s 2 s + 1 10 s + 5 {y} = so that y=− s2 (s 1 3 13 4 1 6 + t + e−t − e−5t − (t − 2) − (t − 2) 25 5 2 50 25 5 1 −(t−2) 9 −5(t−2) + e (t − 2) − e (t − 2). 4 100 (t − 2) 36. Taking the Laplace transform of the differential equation we obtain {y} = s3 + 2 2 + 2s + s2 −s − 3 e − 5) s (s − 5) s3 (s =− . 1 2 127 1 1 2 37 2 1 37 1 12 1 1 2 1 − + − + − − − − e−s 125 s 25 s2 5 s3 125 s − 5 125 s 25 s2 5 s3 125 s − 5 so that . 2 37 37 5(t−1) 2 1 2 127 5t 12 1 2 y=− − t− t + e − − − (t − 1) − (t − 1) + e (t − 1). 125 25 5 125 125 25 5 125 37. Taking the Laplace transform of the integral equation we obtain {y} = so that 1 1 1 2 + + s s2 2 s3 1 y(t) = 1 + t + t2 . 2 38. Taking the Laplace transform of the integral equation we obtain ( {f })2 = 6 · 6 s4 or 248 {f } = ±6 · 1 s2 CHAPTER 4 REVIEW EXERCISES so that f (t) = ±6t. 39. Taking the Laplace transform of the system gives s 4 {x} + {y} = {x} + s 1 +1 s2 {y} = 2 so that {x} = 1 1 1 1 9 1 s2 − 2s + 1 =− + + . s(s − 2)(s + 2) 4 s 8 s−2 8 s+2 Then 1 1 9 x = − + e2t + e−2t 4 8 8 and y = −x′ + t = 9 −2t 1 2t e − e + t. 4 4 40. Taking the Laplace transform of the system gives s2 {x} + s2 2s {x} + s2 1 s−2 1 {y} = − s−2 {y} = so that {x} = 1 1 1 1 1 2 = − + s(s − 2)2 2 s 2 s − 2 (s − 2)2 {y} = 3 1 1 1 3 1 1 −s − 2 =− − + − . s2 (s − 2)2 4 s 2 s2 4 s − 2 (s − 2)2 and Then x= 1 1 2t − e + te2t 2 2 3 1 3 and y = − − t + e2t − te2t . 4 2 4 41. The integral equation is 10i + 2 ! t i(τ ) dτ = 2t2 + 2t. 0 Taking the Laplace transform we obtain # $ 4 2 45 9 s+2 9 2 9 2 s {i} = + = 2 =− + 2 + =− + 2 + . s3 s2 10s + 2 s (5s + 2) s s 5s + 1 s s s + 1/5 Thus i(t) = −9 + 2t + 9e−t/5 . 42. The differential equation is 1 d2 q dq + 10 + 100q = 10 − 10 2 dt2 dt Taking the Laplace transform we obtain {q} = = s(s2 - ; < 20 1 − e−5s + 20s + 200) (t − 5). . < ; 1 1 1 1 s + 10 10 − − 1 − e−5s 2 2 2 2 10 s 10 (s + 10) + 10 10 (s + 10) + 10 249 CHAPTER 4 REVIEW EXERCISES so that q(t) = 1 1 1 − e−10t cos 10t − e−10t sin 10t 10 10 10 . 1 −10(t−5) 1 −10(t−5) 1 − e cos 10(t − 5) − e sin 10(t − 5) (t − 5). − 10 10 10 43. Taking the Laplace transform of the given differential equation we obtain $ # 2w0 c1 2! 1 5! 1 5! c2 3! L 4! {y} = · 5− · 6+ · 6 e−sL/2 + · + · EIL 48 s 120 s 120 s 2 s3 6 s4 so that ) # $5 1 5 1 L 2w0 L 4 y= x − x + x− EIL 48 120 120 2 # L x− 2 $ c2 c1 + x2 + x3 2 6 , where y ′′ (0) = c1 and y ′′′ (0) = c2 . Using y ′′ (L) = 0 and y ′′′ (L) = 0 we find c1 = w0 L2 /24EI, Hence c2 = −w0 L/4EI. ) # $5 1 5 L 4 L2 3 L3 2 1 L w0 − x + x − x + x + x− y= 12EIL 5 2 2 4 5 2 # L x− 2 $, . 44. In this case the boundary conditions are y(0) = y ′ (0) = 0 and y(π) = y ′ (π) = 0. If we let c1 = y ′′ (0) and c2 = y ′′′ (0) then s4 {y} − s3 y(0) − s2 y ′ (0) − sy(0) − y ′′′ (0) + 4 and {y} = {y} = {δ(t − π/2)} c1 2s c2 4 w0 4 · + · + · e−sπ/2 . 2 s4 + 4 4 s4 + 4 4EI s4 + 4 From the table of transforms we get c1 c2 y= sin x sinh x + (sin x cosh x − cos x sinh x) 2 4 0 0 0 π1 π1 π1 π 1? w0 > 0 + sin x − cosh x − − cos x − sinh x − 4EI 2 2 2 2 Using y(π) = 0 and y ′ (π) = 0 we find c1 = Hence y= w0 sinh π2 , EI sinh π c2 = − x− π1 2 x− π1 . 2 0 w0 cosh π2 . EI sinh π w0 sinh π2 w0 cosh π2 sin x sinh x − (sin x cosh x − cos x sinh x) 2EI sinh π 4EI sinh π 0 0 0 π1 π1 π1 π 1? w0 > 0 + sin x − cosh x − − cos x − sinh x − 4EI 2 2 2 2 45. (a) With ω 2 = g/l and K = k/m the system of differential equations is 0 θ1′′ + ω 2 θ1 = −K(θ1 − θ2 ) θ2′′ + ω 2 θ2 = K(θ1 − θ2 ). Denoting the Laplace transform of θ(t) by Θ(s) we have that the Laplace transform of the system is (s2 + ω 2 )Θ1 (s) = −KΘ1 (s) + KΘ2 (s) + sθ0 (s2 + ω 2 )Θ2 (s) = KΘ1 (s) − KΘ2 (s) + sψ0 . 250 CHAPTER 4 REVIEW EXERCISES If we add the two equations, we get Θ1 (s) + Θ2 (s) = (θ0 + ψ0 ) s s2 + ω 2 which implies θ1 (t) + θ2 (t) = (θ0 + ψ0 ) cos ωt. This enables us to solve for first, say, θ1 (t) and then find θ2 (t) from θ2 (t) = −θ1 (t) + (θ0 + ψ0 ) cos ωt. Now solving (s2 + ω 2 + K)Θ1 (s) − KΘ2 (s) = sθ0 −kΘ1 (s) + (s2 + ω 2 + K)Θ2 (s) = sψ0 gives [(s2 + ω 2 + K)2 − K 2 ]Θ1 (s) = s(s2 + ω 2 + K)θ0 + Ksψ0 . Factoring the difference of two squares and using partial fractions we get Θ1 (s) = θ0 + ψ0 θ0 − ψ0 s(s2 + ω 2 + K)θ0 + Ksψ0 s s = + , 2 2 2 2 2 2 2 (s + ω )(s + ω + 2K) 2 s +ω 2 s + ω 2 + 2K so = θ0 + ψ0 θ0 − ψ0 cos ωt + cos ω 2 + 2K t. 2 2 Then from θ2 (t) = −θ1 (t) + (θ0 + ψ0 ) cos ωt we get = θ0 − ψ0 θ0 + ψ0 cos ωt − cos ω 2 + 2K t. θ2 (t) = 2 2 θ1 (t) = (b) With the initial conditions θ1 (0) = θ0 , θ1′ (0) = 0, θ2 (0) = θ0 , θ2′ (0) = 0 we have θ1 (t) = θ0 cos ωt, θ2 (t) = θ0 cos ωt. Physically this means that both pendulums swing in the same direction as if they were free since the spring exerts no influence on the motion (θ1 (t) and θ2 (t) are free of K). With the initial conditions θ1 (0) = θ0 , θ1′ (0) = 0, θ2 (0) = −θ0 , θ2′ (0) = 0 we have = = θ1 (t) = θ0 cos ω 2 + 2K t, θ2 (t) = −θ0 cos ω 2 + 2K t. Physically this means that both pendulums swing in the opposite directions, stretching and compressing the spring. The amplitude of both displacements is |θ0 |. Moreover, θ1 (t) = θ0 and θ2 (t) = −θ0 at precisely the same times. At these times the spring is stretched to its maximum. 251 5 Series Solutions of Linear Differential Equations EXERCISES 5.1 Solutions About Ordinary Points ! ! n+1 n+1 !2 2n x /(n + 1) !! ! = lim |x| = 2|x| 1. lim ! ! n n n→∞ n→∞ 2 x /n n+1 The series is absolutely convergent for 2|x| < 1 or |x| < 21 . The radius of convergence is R = 21 . At x = − 21 , the "∞ "∞ series n=1 (−1)n /n converges by the alternating series test. At x = 21 , the series n=1 1/n is the harmonic series which diverges. Thus, the given series converges on [− 21 , 12 ). ! ! ! 100n+1 (x + 7)n+1 /(n + 1)! ! ! ! = lim 100 |x + 7| = 0 2. lim ! ! n→∞ n + 1 n→∞ 100n (x + 7)n /n! The radius of convergence is R = ∞. The series is absolutely convergent on (−∞, ∞). 3. By the ratio test, ! ! ! (x − 5)n+1 /10n+1 ! ! = lim 1 |x − 5| = 1 |x − 5|. ! lim n→∞ ! (x − 5)n /10n ! n→∞ 10 10 1 |x − 5| < 1, |x − 5| < 10, or on (−5, 15). The radius of convergence is The series is absolutely convergent for 10 "∞ "∞ n R = 10. At x = −5, the series n=1 (−1) (−10)n /10n = n=1 1 diverges by the nth term test. At x = 15, the "∞ "∞ series n=1 (−1)n 10n /10n = n=1 (−1)n diverges by the nth term test. Thus, the series converges on (−5, 15). ! ! # ! (n + 1)!(x − 1)n+1 ! ! = lim (n + 1)|x − 1| = ∞, x ̸= 1 4. lim !! ! n→∞ n→∞ n!(x − 1)n 0, x=1 The radius of convergence is R = 0 and the series converges only for x = 1. %$ % $ x5 x7 x2 x4 x6 2x3 2x5 4x7 x3 + − + ··· 1− + − + ··· = x − + − + ··· 5. sin x cos x = x − 6 120 5040 2 24 720 3 15 315 $ %$ % x3 x4 x2 x4 x3 x4 x2 − + − ··· 1− + − ··· = 1 − x + − + ··· 6. e−x cos x = 1 − x + 2 6 24 2 24 3 6 1 1 5x4 61x6 x2 = + + + ··· =1+ 6 4 2 cos x 2 4! 6! 1 − x2 + x − x + · · · 4! 6! Since cos(π/2) = cos(−π/2) = 0, the series converges on (−π/2, π/2). 1−x 1 3 3 3 = − x + x2 − x3 + · · · 8. 2+x 2 4 8 16 Since the function is undefined at x = −2, the series converges on (−2, 2). 7. 9. Let k = n + 2 so that n = k − 2 and ∞ & n=1 ncn xn+2 = ∞ & k=3 (k − 2)ck−2 xk . 252 5.1 Solutions About Ordinary Points 10. Let k = n − 3 so that n = k + 3 and ∞ & n=3 11. ∞ & 2ncn xn−1 + ∞ & (2n − 1)cn xn−3 = ∞ & 6cn xn+1 = 2 · 1 · c1 x0 + = 2c1 + ∞ & ' 2ncn xn−1 + = 2c1 + ∞ & 6cn xn+1 n=0 () ' * k=n−1 () k=n+1 ∞ & 2(k + 1)ck+1 xk + k=1 ∞ & (2k + 5)ck+3 xk . k=0 n=2 n=0 n=1 ∞ & 6ck−1 xk * k=1 [2(k + 1)ck+1 + 6ck−1 ]xk k=1 12. ∞ & n=2 n(n − 1)cn xn + 2 ∞ & n=2 0 n(n − 1)cn xn−2 + 3 1 1 = 2 · 2 · 1c2 x + 2 · 3 · 2c3 x + 3 · 1 · c1 x + ∞ & n=1 ∞ & n=2 = 4c2 + (3c1 + 12c3 )x + = 4c2 + (3c1 + 12c3 )x + = 4c2 + (3c1 + 12c3 )x + ∞ & k=2 ∞ & k=2 ∞ & k=2 13. y ′ = ∞ & (−1)n+1 xn−1 , ' k(k − 1)ck xk + 2 n=2 = n=2 = = ∞ & k=1 ∞ & k=1 14. y ′ = () * k=n ∞ & n=4 n−2 n(n − 1)cn x ' () k=2 ncn xn ' * kck xk () k=n * k=2 (−1)n+1 (n − 1)xn−2 (−1)n+1 (n − 1)xn−1 + ∞ & n=2 ∞ & n=2 ∞ & (−1)n+1 xn−1 n=1 (−1)n+1 (n − 1)xn−2 + (−1)n+1 (n − 1)xn−1 + ' (−1)k+2 kxk + () k=n−1 ∞ & * ∞ & n=3 ' (−1)k+3 (k + 1)xk + k=1 ∞ & y ′′ = (−1)n+1 (n − 1)xn−2 + () k=n−2 ∞ & (−1)k+2 xk k=1 ∞ & (−1)n 2n(2n − 1) 2n−2 x 22n (n!)2 n=1 253 (−1)n+1 xn−1 n=1 + . (−1)k+2 k − (−1)k+2 k − (−1)k+2 + (−1)k+2 xk = 0 (−1)n 2n 2n−1 x , 22n (n!)2 n=1 ∞ & +3 ∞ & n=2 k=n−2 (k + 2)(k + 1)ck+2 xk + 3 (−1)n+1 (n − 1)xn−2 + = −x0 + x0 + ∞ & ∞ & n=2 ∞ & n(n − 1)cn x +2 ∞ & + . k(k + 2)ck + 2(k + 1)(k + 2)ck+2 xk n=1 (x + 1)y ′′ + y ′ = (x + 1) n +, . k(k − 1) + 3k ck + 2(k + 2)(k + 1)ck+2 xk y ′′ = ∞ & ncn xn * ∞ & (−1)n+1 xn−1 n=2 ' () k=n−1 * 5.1 Solutions About Ordinary Points xy ′′ + y ′ + xy = ∞ ∞ ∞ & (−1)n 2n(2n − 1) 2n−1 & (−1)n 2n 2n−1 & (−1)n 2n+1 + + x x x 22n (n!)2 22n (n!)2 22n (n!)2 n=1 n=0 n=1 () * ' () * ' () * ' k=n ∞ / & k=n k k=n+1 k k−1 0 (−1) 2k(2k − 1) (−1) 2k (−1) x2k−1 + 2k + 2k−2 2k 2 2 2 (k!) 2 (k!) 2 [(k − 1)!]2 k=1 0 ∞ / & (−1)k (2k)2 (−1)k = − 2k−2 x2k−1 22k (k!)2 2 [(k − 1)!]2 k=1 0 / ∞ & (2k)2 − 22 k 2 2k−1 = (−1)k x =0 22k (k!)2 = k=1 15. The singular points of (x2 − 25)y ′′ + 2xy ′ + y = 0 are −5 and 5. The distance from 0 to either of these points is 5. The distance from 1 to the closest of these points is 4. 16. The singular points of (x2 − 2x + 10)y ′′ + xy ′ − 4y = 0 are 1 + 3i and 1 − 3i. The distance from 0 to either of √ these points is 10 . The distance from 1 to either of these points is 3. "∞ 17. Substituting y = n=0 cn xn into the differential equation we have y ′′ − xy = ∞ & n=2 ' n(n − 1)cn xn−2 − = 2c2 + () k=n−2 ∞ & k=1 * ∞ & cn xn+1 = n=0 ' () k=n+1 * ∞ & k=0 (k + 2)(k + 1)ck+2 xk − ∞ & ck−1 xk k=1 [(k + 2)(k + 1)ck+2 − ck−1 ]xk = 0. Thus c2 = 0 (k + 2)(k + 1)ck+2 − ck−1 = 0 and ck+2 = 1 ck−1 , (k + 2)(k + 1) k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 1 6 c4 = c5 = 0 1 c6 = 180 c3 = and so on. For c0 = 0 and c1 = 1 we obtain c3 = 0 1 c4 = 12 c5 = c6 = 0 1 c7 = 504 and so on. Thus, two solutions are 1 6 1 x + ··· y1 = 1 + x3 + 6 180 and 254 y2 = x + 1 4 1 7 x + x + ··· . 12 504 5.1 18. Substituting y = ′′ "∞ n=0 2 y +x y = Solutions About Ordinary Points cn xn into the differential equation we have ∞ & n−2 n=2 ' n(n − 1)cn x () n+2 cn x = n=0 k=n−2 ∞ & = 2c2 + 6c3 x + + ∞ & * ' () * k=n+2 ∞ & k (k + 2)(k + 1)ck+2 x + k=0 ∞ & ck−2 xk k=2 [(k + 2)(k + 1)ck+2 + ck−2 ]xk = 0. k=2 Thus c2 = c3 = 0 (k + 2)(k + 1)ck+2 + ck−2 = 0 and ck+2 = − 1 ck−2 , (k + 2)(k + 1) k = 2, 3, 4, . . . . Choosing c0 = 1 and c1 = 0 we find 1 12 c5 = c6 = c7 = 0 1 c8 = 672 c4 = − and so on. For c0 = 0 and c1 = 1 we obtain c4 = 0 1 20 c6 = c7 = c8 = 0 1 c9 = 1440 c5 = − and so on. Thus, two solutions are y1 = 1 − 19. Substituting y = "∞ n=0 1 8 1 4 x + x − ··· 12 672 and y2 = x − 1 5 1 9 x + x − ··· . 20 1440 cn xn into the differential equation we have y ′′ − 2xy ′ + y = = ∞ & n=2 ' ∞ & k=0 n(n − 1)cn xn−2 − 2 () * k=n−2 ∞ & ncn xn + n=1 ' = 2c2 + c0 + k=1 cn xn n=0 () k=n (k + 2)(k + 1)ck+2 xk − 2 ∞ & ∞ & ∞ & * ' () * k=n kck xk + k=1 ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 − (2k − 1)ck ]xk = 0. Thus 2c2 + c0 = 0 (k + 2)(k + 1)ck+2 − (2k − 1)ck = 0 255 5.1 Solutions About Ordinary Points and 1 c2 = − c0 2 2k − 1 ck+2 = ck , (k + 2)(k + 1) k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 1 2 c3 = c5 = c7 = · · · = 0 1 c4 = − 8 7 c6 = − 240 c2 = − and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 1 c3 = 6 1 c5 = 24 1 c7 = 112 and so on. Thus, two solutions are 1 1 7 6 y1 = 1 − x2 − x4 − x − ··· 2 8 240 20. Substituting y = "∞ n=0 1 1 1 7 y2 = x + x3 + x5 + x + ··· . 6 24 112 and cn xn into the differential equation we have y ′′ − xy ′ + 2y = = ∞ & n=2 ' n(n − 1)cn xn−2 − () k=n−2 ∞ & k=0 * ∞ & ncn xn + 2 n=1 ' = 2c2 + 2c0 + k=1 k=n ∞ & * ' () * k=n kck xk + 2 k=1 ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 − (k − 2)ck ]xk = 0. Thus 2c2 + 2c0 = 0 (k + 2)(k + 1)ck+2 − (k − 2)ck = 0 and c2 = −c0 ck+2 = cn xn n=0 () (k + 2)(k + 1)ck+2 xk − ∞ & ∞ & k−2 ck , (k + 2)(k + 1) 256 k = 1, 2, 3, . . . . 5.1 Solutions About Ordinary Points Choosing c0 = 1 and c1 = 0 we find c2 = −1 c3 = c5 = c7 = · · · = 0 c4 = 0 c6 = c8 = c10 = · · · = 0. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 1 c3 = − 6 1 c5 = − 120 and so on. Thus, two solutions are y1 = 1 − x2 21. Substituting y = "∞ n=0 and 1 5 1 x − ··· . y2 = x − x3 − 6 120 cn xn into the differential equation we have y ′′ + x2 y ′ + xy = ∞ & n=2 = ' ∞ & n(n − 1)cn xn−2 + () k=n−2 ∞ & n=1 ' * ncn xn+1 + k=0 ∞ & k=2 = 2c2 + (6c3 + c0 )x + ∞ & cn xn+1 n=0 () k=n+1 (k + 2)(k + 1)ck+2 xk + ∞ & * ' () k=n+1 (k − 1)ck−1 xk + Thus c2 = 0 6c3 + c0 = 0 (k + 2)(k + 1)ck+2 + kck−1 = 0 and c2 = 0 1 c3 = − c0 6 k ck−1 , (k + 2)(k + 1) Choosing c0 = 1 and c1 = 0 we find 1 6 c4 = c5 = 0 1 c6 = 45 c3 = − 257 ∞ & ck−1 xk k=1 [(k + 2)(k + 1)ck+2 + kck−1 ]xk = 0. k=2 ck+2 = − * k = 2, 3, 4, . . . . 5.1 Solutions About Ordinary Points and so on. For c0 = 0 and c1 = 1 we obtain c3 = 0 1 6 c5 = c6 = 0 5 c7 = 252 c4 = − and so on. Thus, two solutions are 1 1 y1 = 1 − x3 + x6 − · · · 6 45 22. Substituting y = "∞ n=0 1 5 7 y2 = x − x4 + x − ··· . 6 252 and cn xn into the differential equation we have y ′′ + 2xy ′ + 2y = ∞ & n=2 = ' ∞ & n(n − 1)cn xn−2 + 2 () * k=n−2 ∞ & ncn xn + 2 n=1 ' k=n ∞ & * = 2c2 + 2c0 + Thus 2c2 + 2c0 = 0 (k + 2)(k + 1)ck+2 + 2(k + 1)ck = 0 and c2 = −c0 2 ck , k+2 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find c2 = −1 c3 = c5 = c7 = · · · = 0 c4 = 1 2 c6 = − k=n ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 + 2(k + 1)ck ]xk = 0. k=1 ck+2 = − ' () * kck xk + 2 k=1 ∞ & cn xn n=0 () (k + 2)(k + 1)ck+2 xk + 2 k=0 ∞ & 1 6 and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 2 c3 = − 3 4 c5 = 15 8 c7 = − 105 258 5.1 Solutions About Ordinary Points and so on. Thus, two solutions are 1 1 2 4 8 7 y1 = 1 − x2 + x4 − x6 + · · · and y2 = x − x3 + x5 − x + ··· . 2 6 3 15 105 "∞ 23. Substituting y = n=0 cn xn into the differential equation we have (x − 1)y ′′ + y ′ = = ∞ & n=2 ' n(n − 1)cn xn−1 − () * k=n−1 ∞ & k=1 (k + 1)kck+1 xk − = −2c2 + c1 + Thus ∞ & k=1 ∞ & n=2 ' ∞ & n(n − 1)cn xn−2 + () * k=n−2 ∞ & ncn xn−1 n=1 ' () * k=n−1 (k + 2)(k + 1)ck+2 xk + k=0 ∞ & (k + 1)ck+1 xk k=0 [(k + 1)kck+1 − (k + 2)(k + 1)ck+2 + (k + 1)ck+1 ]xk = 0. −2c2 + c1 = 0 (k + 1)2 ck+1 − (k + 2)(k + 1)ck+2 = 0 and 1 c1 2 k+1 ck+1 , = k+2 c2 = ck+2 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find c2 = c3 = c4 = · · · = 0. For c0 = 0 and c1 = 1 we obtain 1 1 1 c2 = , c3 = , c4 = , 2 3 4 and so on. Thus, two solutions are 1 1 1 y1 = 1 and y2 = x + x2 + x3 + x4 + · · · . 2 3 4 "∞ 24. Substituting y = n=0 cn xn into the differential equation we have (x + 2)y ′′ + xy ′ − y = = ∞ & n=2 ' n(n − 1)cn xn−1 + () k=n−1 ∞ & * (k + 1)kck+1 xk + k=1 ∞ & n=2 ' ∞ & 2n(n − 1)cn xn−2 + () k=n−2 * ∞ & n=1 ' ncn xn − () k=n 2(k + 2)(k + 1)ck+2 xk + k=0 ∞ & k=1 * ∞ & cn xn n=0 ' () * k=n kck xk − ∞ & ck xk k=0 ∞ & . + = 4c2 − c0 + (k + 1)kck+1 + 2(k + 2)(k + 1)ck+2 + (k − 1)ck xk = 0. k=1 Thus 4c2 − c0 = 0 (k + 1)kck+1 + 2(k + 2)(k + 1)ck+2 + (k − 1)ck = 0, and 1 c0 4 (k + 1)kck+1 + (k − 1)ck =− , 2(k + 2)(k + 1) k = 1, 2, 3, . . . c2 = ck+2 259 k = 1, 2, 3, . . . . 5.1 Solutions About Ordinary Points Choosing c0 = 1 and c1 = 0 we find c1 = 0, c2 = 1 , 4 c3 = − 1 , 24 c4 = 0, c5 = 1 480 and so on. For c0 = 0 and c1 = 1 we obtain c2 = 0 c3 = 0 c4 = c5 = c6 = · · · = 0. Thus, two solutions are 0 / 1 1 5 1 y1 = c0 1 + x2 − x3 + x + ··· 4 24 480 25. Substituting y = "∞ n=0 and y2 = c1 x. cn xn into the differential equation we have y ′′ − (x + 1)y ′ − y = = ∞ & n=2 ' ∞ & k=0 n(n − 1)cn xn−2 − () * k=n−2 ∞ & n=1 ' ncn xn − () k=n (k + 2)(k + 1)ck+2 xk − = 2c2 − c1 − c0 + ∞ & k=1 ∞ & k=1 * ∞ & n=1 ' ncn xn−1 − () k=n−1 kck xk − ∞ & k=0 * ∞ & cn xn n=0 ' () * k=n (k + 1)ck+1 xk − ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 − (k + 1)ck+1 − (k + 1)ck ]xk = 0. Thus 2c2 − c1 − c0 = 0 (k + 2)(k + 1)ck+2 − (k + 1)(ck+1 + ck ) = 0 and c1 + c0 2 ck+1 + ck = , k+2 c2 = ck+2 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find c2 = 1 , 2 c3 = 1 , 6 c4 = 1 , 6 1 , 2 c3 = 1 , 2 c4 = 1 , 4 and so on. For c0 = 0 and c1 = 1 we obtain c2 = and so on. Thus, two solutions are 1 1 1 y1 = 1 + x2 + x3 + x4 + · · · 2 6 6 and 260 1 1 1 y2 = x + x2 + x3 + x4 + · · · . 2 2 4 5.1 26. Substituting y = , "∞ n=0 Solutions About Ordinary Points cn xn into the differential equation we have ∞ ∞ ∞ & & & n(n − 1)cn xn−2 − 6 cn xn n(n − 1)cn xn + x2 + 1 y ′′ − 6y = n=2 n=2 = ' ∞ & k=2 () * k=n k(k − 1)ck xk + n=0 () ' k=n−2 ∞ & k=0 * ' () * k=n (k + 2)(k + 1)ck+2 xk − 6 ∞ & ck xk k=0 ∞ & +, 2 . = 2c2 − 6c0 + (6c3 − 6c1 )x + k − k − 6 ck + (k + 2)(k + 1)ck+2 xk = 0. k=2 Thus 2c2 − 6c0 = 0 6c3 − 6c1 = 0 (k − 3)(k + 2)ck + (k + 2)(k + 1)ck+2 = 0 and c2 = 3c0 c3 = c1 ck+2 = − k−3 ck , k+1 k = 2, 3, 4, . . . . Choosing c0 = 1 and c1 = 0 we find c2 = 3 c3 = c5 = c7 = · · · = 0 c4 = 1 c6 = − 1 5 and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 c3 = 1 c5 = c7 = c9 = · · · = 0. Thus, two solutions are 1 y1 = 1 + 3x2 + x4 − x6 + · · · 5 27. Substituting y = , "∞ n=0 and y2 = x + x3 . cn xn into the differential equation we have ∞ ∞ ∞ ∞ & & & & x2 + 2 y ′′ + 3xy ′ − y = n(n − 1)cn xn−2 + 3 ncn xn − cn xn n(n − 1)cn xn + 2 n=2 n=2 = ∞ & k=2 ' () k=n k(k − 1)ck xk + 2 ∞ & * ' n=1 () k=n−2 (k + 2)(k + 1)ck+2 xk + 3 k=0 = (4c2 − c0 ) + (12c3 + 2c1 )x + ∞ & k=1 ∞ & k=2 * ' kck xk − n=0 () k=n ∞ & * ck xk ' () * k=n k=0 + , - . 2(k + 2)(k + 1)ck+2 + k 2 + 2k − 1 ck xk = 0. 261 5.1 Solutions About Ordinary Points Thus 4c2 − c0 = 0 12c3 + 2c1 = 0 , 2(k + 2)(k + 1)ck+2 + k 2 + 2k − 1 ck = 0 and 1 c0 4 1 c3 = − c1 6 k 2 + 2k − 1 ck+2 = − ck , 2(k + 2)(k + 1) c2 = k = 2, 3, 4, . . . . Choosing c0 = 1 and c1 = 0 we find 1 4 c3 = c5 = c7 = · · · = 0 7 c4 = − 96 c2 = and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 1 c3 = − 6 7 c5 = 120 and so on. Thus, two solutions are 7 1 y1 = 1 + x2 − x4 + · · · 4 96 28. Substituting y = , "∞ n=0 1 7 5 y2 = x − x3 + x − ··· . 6 120 and cn xn into the differential equation we have ∞ ∞ ∞ ∞ & & & & n(n − 1)cn xn − x2 − 1 y ′′ + xy ′ − y = n(n − 1)cn xn−2 + ncn xn − cn xn n=2 = ' ∞ & k=2 n=2 () k=n * k(k − 1)ck xk − ' n=1 () k=n−2 ∞ & * ' k=n (k + 2)(k + 1)ck+2 xk + k=0 = (−2c2 − c0 ) − 6c3 x + n=0 () ∞ & k=2 k=1 ' () * k=n kck xk − ∞ & ck xk k=0 + , - . −(k + 2)(k + 1)ck+2 + k 2 − 1 ck xk = 0. Thus −2c2 − c0 = 0 −6c3 = 0 −(k + 2)(k + 1)ck+2 + (k − 1)(k + 1)ck = 0 262 ∞ & * 5.1 and Solutions About Ordinary Points 1 c2 = − c0 2 c3 = 0 ck+2 = Choosing c0 = 1 and c1 = 0 we find k−1 ck , k+2 k = 2, 3, 4, . . . . 1 2 c3 = c5 = c7 = · · · = 0 1 c4 = − 8 c2 = − and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 c3 = c5 = c7 = · · · = 0. Thus, two solutions are 1 1 y1 = 1 − x2 − x4 − · · · 2 8 29. Substituting y = "∞ n=0 and y2 = x. cn xn into the differential equation we have (x − 1)y ′′ − xy ′ + y = = ∞ & k=1 ∞ & n=2 n(n − 1)cn xn−1 − ' () k=n−1 (k + 1)kck+1 xk − = −2c2 + c0 + ∞ & k=1 ∞ & k=0 * ∞ & n=2 n(n − 1)cn xn−2 − ' () k=n−2 (k + 2)(k + 1)ck+2 xk − * ∞ & ∞ & n=1 ' kck xk + k=1 cn xn * ∞ & ' () * k=n ck xk k=0 [−(k + 2)(k + 1)ck+2 + (k + 1)kck+1 − (k − 1)ck ]xk = 0. −2c2 + c0 = 0 −(k + 2)(k + 1)ck+2 + (k − 1)kck+1 − (k − 1)ck = 0 1 c0 2 (k − 1)ck kck+1 = − , k+2 (k + 2)(k + 1) c2 = ck+2 ∞ & n=0 () k=n Thus and ncn xn + k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 1 1 , c3 = , c4 = 0, 2 6 and so on. For c0 = 0 and c1 = 1 we obtain c2 = c3 = c4 = · · · = 0. Thus, $ % 1 2 1 3 y = C1 1 + x + x + · · · + C2 x 2 6 and % $ 1 y ′ = C1 x + x2 + · · · + C2 . 2 c2 = 263 5.1 Solutions About Ordinary Points The initial conditions imply C1 = −2 and C2 = 6, so % $ 1 1 y = −2 1 + x2 + x3 + · · · + 6x = 8x − 2ex . 2 6 "∞ 30. Substituting y = n=0 cn xn into the differential equation we have (x+1)y ′′ − (2 − x)y ′ + y = ∞ & n=2 = ' ∞ & n(n − 1)cn xn−1 + () k=n−1 * (k + 1)kck+1 xk + k=1 ∞ & n=2 ' = 2c2 − 2c1 + c0 + k=1 () k=n−2 ∞ & k=0 ∞ & n(n − 1)cn xn−2 − 2 * ∞ & ncn xn−1 + n=1 ' ncn xn + n=1 () k=n−1 (k + 2)(k + 1)ck+2 xk − 2 ∞ & ∞ & * ' ∞ & n=0 () k=n (k + 1)ck+1 xk + k=0 * ∞ & ' () * k=n kck xk + k=1 [(k + 2)(k + 1)ck+2 − (k + 1)ck+1 + (k + 1)ck ]xk = 0. Thus 2c2 − 2c1 + c0 = 0 (k + 2)(k + 1)ck+2 − (k + 1)ck+1 + (k + 1)ck = 0 and 1 c2 = c1 − c0 2 1 1 ck+1 − ck , ck+2 = k+2 k+2 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 1 c2 = − , 2 1 c3 = − , 6 c4 = 1 , 12 and so on. For c0 = 0 and c1 = 1 we obtain c2 = 1, c3 = 0, 1 c4 = − , 4 and so on. Thus, and % % $ $ 1 1 1 1 y = C1 1 − x2 − x3 + x4 + · · · + C2 x + x2 − x4 + · · · 2 6 12 4 % $ , 1 1 y ′ = C1 −x − x2 + x3 + · · · + C2 1 + 2x − x3 + · · · . 2 3 The initial conditions imply C1 = 2 and C2 = −1, so $ % $ % 1 2 1 3 1 4 1 4 2 y = 2 1 − x − x + x + ··· − x + x − x + ··· 2 6 12 4 1 5 = 2 − x − 2x2 − x3 + x4 + · · · . 3 12 264 cn xn ∞ & k=0 ck xk 5.1 31. Substituting y = "∞ n=0 Solutions About Ordinary Points cn xn into the differential equation we have y ′′ − 2xy ′ + 8y = = ∞ & n=2 ' n(n − 1)cn xn−2 − 2 () k=0 ncn xn + 8 n=1 ' * k=n−2 ∞ & ∞ & = 2c2 + 8c0 + k=1 cn xn n=0 () k=n (k + 2)(k + 1)ck+2 xk − 2 ∞ & ∞ & ∞ & * ' () * k=n kck xk + 8 k=1 ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 + (8 − 2k)ck ]xk = 0. Thus 2c2 + 8c0 = 0 (k + 2)(k + 1)ck+2 + (8 − 2k)ck = 0 and c2 = −4c0 ck+2 = 2(k − 4) ck , (k + 2)(k + 1) k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find c2 = −4 c3 = c5 = c7 = · · · = 0 4 c4 = 3 c6 = c8 = c10 = · · · = 0. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 c3 = −1 1 c5 = 10 and so on. Thus, and $ % $ % 1 4 y = C1 1 − 4x2 + x4 + C2 x − x3 + x5 + · · · 3 10 ′ y = C1 $ 16 −8x + x3 3 % + C2 $ % 1 4 1 − 3x + x + · · · . 2 2 The initial conditions imply C1 = 3 and C2 = 0, so % $ 4 y = 3 1 − 4x2 + x4 = 3 − 12x2 + 4x4 . 3 265 5.1 Solutions About Ordinary Points 32. Substituting y = "∞ n=0 cn xn into the differential equation we have (x2 + 1)y ′′ + 2xy ′ = ∞ & n=2 = ' ∞ & k=2 n(n − 1)cn xn + () * k=n k(k − 1)ck xk + ∞ & n=2 ' n(n − 1)cn xn−2 + ∞ & * 2ncn xn n=1 ' k=0 ∞ & k=2 () k=n (k + 2)(k + 1)ck+2 xk + = 2c2 + (6c3 + 2c1 )x + Thus () k=n−2 ∞ & ∞ & * 2kck xk k=1 + . k(k + 1)ck + (k + 2)(k + 1)ck+2 xk = 0. 2c2 = 0 6c3 + 2c1 = 0 k(k + 1)ck + (k + 2)(k + 1)ck+2 = 0 and c2 = 0 1 c3 = − c1 3 k ck , ck+2 = − k+2 k = 2, 3, 4, . . . . Choosing c0 = 1 and c1 = 0 we find c3 = c4 = c5 = · · · = 0. For c0 = 0 and c1 = 1 we obtain 1 3 c4 = c6 = c8 = · · · = 0 1 c5 = − 5 1 c7 = 7 c3 = − and so on. Thus y = C0 + C1 and $ % 1 3 1 5 1 7 x − x + x − x + ··· 3 5 7 , y ′ = c1 1 − x2 + x4 − x6 + · · · . The initial conditions imply c0 = 0 and c1 = 1, so 1 1 1 y = x − x3 + x5 − x7 + · · · . 3 5 7 "∞ cn xn into the differential equation we have $ % ∞ & , 1 1 5 n(n − 1)cn xn−2 + x − x3 + y ′′ + (sin x)y = x − · · · c0 + c1 x + c2 x2 + · · · 6 120 n=2 / $ % 0 + . 1 2 3 2 3 = 2c2 + 6c3 x + 12c4 x + 20c5 x + · · · + c0 x + c1 x + c2 − c0 x + · · · 6 $ % 1 = 2c2 + (6c3 + c0 )x + (12c4 + c1 )x2 + 20c5 + c2 − c0 x3 + · · · = 0. 6 33. Substituting y = n=0 266 5.1 Solutions About Ordinary Points Thus 2c2 = 0 6c3 + c0 = 0 12c4 + c1 = 0 1 20c5 + c2 − c0 = 0 6 c2 = 0 and 1 c3 = − c0 6 1 c4 = − c1 12 1 1 c0 . c5 = − c2 + 20 120 Choosing c0 = 1 and c1 = 0 we find 1 c3 = − , 6 c2 = 0, c4 = 0, c5 = 1 120 and so on. For c0 = 0 and c1 = 1 we obtain c2 = 0, c3 = 0, c4 = − 1 , 12 c5 = 0 and so on. Thus, two solutions are 1 5 1 y1 = 1 − x3 + x + ··· 6 120 34. Substituting y = "∞ n=0 and y2 = x − 1 4 x + ··· . 12 cn xn into the differential equation we have y ′′ + ex y ′ − y = ∞ & n=2 n(n − 1)cn xn−2 % $ ∞ , - & 1 1 cn xn + 1 + x + x2 + x3 + · · · c1 + 2c2 x + 3c3 x2 + 4c4 x3 + · · · − 2 6 n=0 + . 2 3 = 2c2 + 6c3 x + 12c4 x + 20c5 x + · · · / $ % 0 1 2 + c1 + (2c2 + c1 )x + 3c3 + 2c2 + c1 x + · · · − [c0 + c1 x + c2 x2 + · · ·] 2 % $ 1 = (2c2 + c1 − c0 ) + (6c3 + 2c2 )x + 12c4 + 3c3 + c2 + c1 x2 + · · · = 0. 2 Thus 2c2 + c1 − c0 = 0 6c3 + 2c2 = 0 1 12c4 + 3c3 + c2 + c1 = 0 2 267 5.1 Solutions About Ordinary Points and 1 1 c0 − c1 2 2 1 c3 = − c2 3 1 1 1 c4 = − c3 + c2 − c1 . 4 12 24 c2 = Choosing c0 = 1 and c1 = 0 we find c2 = 1 c3 = − , 6 1 , 2 c4 = 0 and so on. For c0 = 0 and c1 = 1 we obtain 1 c2 = − , 2 c3 = 1 , 6 c4 = − 1 24 and so on. Thus, two solutions are 1 1 y1 = 1 + x2 − x3 + · · · 2 6 1 1 1 y2 = x − x2 + x3 − x4 + · · · . 2 6 24 and 35. The singular points of (cos x)y ′′ + y ′ + 5y = 0 are odd integer multiples of π/2. The distance from 0 to either ±π/2 is π/2. The singular point closest to 1 is π/2. The distance from 1 to the closest singular point is then π/2 − 1. "∞ 36. Substituting y = n=0 cn xn into the first differential equation leads to y ′′ − xy = ∞ & n=2 ' n(n − 1)cn xn−2 − = 2c2 + () * k=n−2 ∞ & k=1 ∞ & cn xn+1 = n=0 ' () k=n+1 * ∞ & k=0 (k + 2)(k + 1)ck+2 xk − [(k + 2)(k + 1)ck+2 − ck−1 ]xk = 1. Thus 2c2 = 1 (k + 2)(k + 1)ck+2 − ck−1 = 0 and c2 = ck+2 = 1 2 ck−1 , (k + 2)(k + 1) k = 1, 2, 3, . . . . Let c0 and c1 be arbitrary and iterate to find 1 2 1 c3 = c0 6 1 c4 = c1 12 1 1 c2 = c5 = 20 40 c2 = 268 ∞ & k=1 ck−1 xk 5.1 Solutions About Ordinary Points and so on. The solution is 1 1 1 1 y = c0 + c1 x + x2 + c0 x3 + c1 x4 + c5 + · · · 2 6 12 40 $ % % $ 1 3 1 1 1 4 = c0 1 + x + · · · + c1 x + x + · · · + x2 + x5 + · · · . 6 12 2 40 Substituting y = "∞ n=0 cn xn into the second differential equation leads to y ′′ − 4xy ′ − 4y = = ∞ & n=2 ' ∞ & k=0 n(n − 1)cn xn−2 − () * k=n−2 ∞ & n=1 ' 4ncn xn − k=n (k + 2)(k + 1)ck+2 xk − = 2c2 − 4c0 + ∞ & k=1 () ∞ & k=1 * ∞ & 4cn xn n=0 ' () k=n 4kck xk − ∞ & 4ck xk k=0 + . (k + 2)(k + 1)ck+2 − 4(k + 1)ck xk ∞ & 1 k x x . =e =1+ k! k=1 Thus 2c2 − 4c0 = 1 (k + 2)(k + 1)ck+2 − 4(k + 1)ck = 1 k! and 1 + 2c0 2 1 4 = + ck , (k + 2)! k + 2 c2 = ck+2 * k = 1, 2, 3, . . . . Let c0 and c1 be arbitrary and iterate to find 1 + 2c0 2 1 4 1 4 c3 = + c1 = + c1 3! 3 3! 3 c2 = c4 = 1 4 1 1 13 + c2 = + + 2c0 = + 2c0 4! 4 4! 2 4! c5 = 4 1 4 16 17 16 1 + c3 = + + c1 = + c1 5! 5 5! 5 · 3! 15 5! 15 c6 = 1 1 261 4 4 4 · 13 8 + c4 = + + c0 = + c0 6! 6 6! 6 · 4! 6 6! 3 c7 = 4 1 4 · 17 64 409 64 1 + c5 = + + c1 = + c1 7! 7 7! 7 · 5! 105 7! 105 and so on. The solution is 269 5.1 Solutions About Ordinary Points $ % $ % $ % $ % 1 1 4 13 17 16 2 3 4 y = c0 + c1 x + + 2c0 x + + c1 x + + 2c0 x + + c1 x5 2 3! 3 4! 5! 15 % $ % $ 409 64 261 4 + c0 x6 + + c1 x7 + · · · + 6! 3 7! 105 / / 0 0 4 6 4 3 16 5 64 7 2 4 = c0 1 + 2x + 2x + x + · · · + c1 x + x + x + x + ··· 3 3 15 105 1 1 13 17 261 6 409 7 + x2 + x3 + x4 + x5 + x + x + ··· . 2 3! 4! 5! 6! 7! 37. We identify P (x) = 0 and Q(x) = sin x/x. The Taylor series representation for sin x/x is 1 − x2 /3! + x4 /5! − · · · , for |x| < ∞. Thus, Q(x) is analytic at x = 0 and x = 0 is an ordinary point of the differential equation. 38. If x > 0 and y > 0, then y ′′ = −xy < 0 and the graph of a solution curve is concave down. Thus, whatever portion of a solution curve lies in the first quadrant is concave down. When x > 0 and y < 0, y ′′ = −xy > 0, so whatever portion of a solution curve lies in the fourth quadrant is concave up. "∞ 39. (a) Substituting y = n=0 cn xn into the differential equation we have y ′′ + xy ′ + y = ∞ & n=2 = ' n(n − 1)cn xn−2 + () * k=n−2 ∞ & ∞ & ncn xn + n=1 ' ∞ & n=0 () k=n (k + 2)(k + 1)ck+2 xk + k=0 ∞ & * ' () * k=n kck xk + k=1 = (2c2 + c0 ) + ∞ & k=1 ∞ & ck xk k=0 + . (k + 2)(k + 1)ck+2 + (k + 1)ck xk = 0. Thus 2c2 + c0 = 0 (k + 2)(k + 1)ck+2 + (k + 1)ck = 0 and cn xn 1 c2 = − c0 2 1 ck+2 = − ck , k+2 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 1 2 c3 = c5 = c7 = · · · = 0 11 12 1 c4 = − − = 2 4 2 2 ·2 11 1 2 1 c6 = − =− 3 6 22 · 2 2 · 3! c2 = − and so on. For c0 = 0 and c1 = 1 we obtain 270 5.1 c2 = c4 = c6 = · · · = 0 2 1 c3 = − = − 3 3! 11 12 1 4·2 c5 = − − = = 5 3 5·3 5! 114 · 22 6·4·2 c7 = − =− 7 5! 7! and so on. Thus, two solutions are y1 = ∞ & (−1)k k=0 Solutions About Ordinary Points 2k · k! x2k and y2 = ∞ & (−1)k 2k k! k=0 (2k + 1)! x2k+1 . (b) For y1 , S3 = S2 and S5 = S4 , so we plot S2 , S4 , S6 , S8 , and S10 . y y 4 2 -4 -2 -2 y 4 N=2 4 2 x 2 4 -4 -2 -2 -4 2 2 4 x -4 -2 -2 N=4 -4 N=6 2 4 x y y 4 4 2 2 -4 -2 -2 -4 2 4 x N=8 -4 -2 -2 -4 -4 y y N=10 2 4 x 2 4 x For y2 , S3 = S4 and S5 = S6 , so we plot S2 , S4 , S6 , S8 , and S10 . y y 4 4 2 -4 -2 -2 2 x 2 4 N=2 4 N=4 -4 -2 -2 -4 2 4 -4 -2 -2 y2 4 4 2 2 2 4 2 2 4 x N=6 -4 -2 -2 -4 y1 -2 4 2 x -4 (c) -4 y x -4 -2 -4 2 -2 -2 -4 -4 4 4 N=8 2 4 2 x -4 -2 -2 N=10 -4 x The graphs of y1 and y2 obtained from a numerical solver are shown. We see that the partial sum representations indicate the even and odd natures of the solution, but don’t really give a very accurate representation of the true solution. Increasing N to about 20 gives a much more accurate representation on [−4, 4]. "∞ k "∞ "∞ −x2 /2 2 k k 2k k (d) From ex = = k=0 x /k! we see that e k=0 (−x /2) /k! = k=0 (−1) x /2 k! . From (5) of Section 3.2 we have 271 5.1 Solutions About Ordinary Points 4 3 −x2 /2 3 2 2 e e−x /2 −x2 /2 −x2 /2 dx = e dx = e ex /2 dx y12 e−x2 (e−x2 /2 )2 65 ∞ 3 5∞ 6 3 ∞ ∞ & & (−1)k & 1 (−1)k 2k & 1 2k 2k 2k x x dx = x x dx = 2k k! 2k k! 2k k! 2k k! k=0 k=0 k=0 k=0 5∞ 65 ∞ 6 & (−1)k & 1 2k 2k+1 = x x 2k k! (2k + 1)2k k! y2 = y1 3 e− x dx 2 dx = e−x k=0 /2 3 k=0 1 21 2 1 1 1 1 3 1 1 = 1 − x2 + 2 x4 − 3 x6 + · · · x + x + x5 + x7 + · · · 2 3 2 2 ·2 2 · 3! 3·2 5·2 ·2 7 · 2 · 3! ∞ k k & (−1) 2 k! 2 4·2 5 6·4·2 7 = x − x3 + x − x + ··· = x2k+1 . 3! 5! 7! (2k + 1)! k=0 40. (a) We have y ′′ + (cos x)y = 2c2 + 6c3 x + 12c4 x2 + 20c5 x3 + 30c6 x4 + 42c7 x5 + · · · 2 1 x4 x6 x2 + − + · · · (c0 + c1 x + c2 x2 + c3 x3 + c4 x4 + c5 x5 + · · · ) + 1− 2! 4! 6! 1 1 1 2 1 2 = (2c2 + c0 ) + (6c3 + c1 )x + 12c4 + c2 − c0 x2 + 20c5 + c3 − c1 x3 2 2 1 1 2 4 1 1 2 5 1 1 + 30c6 + c4 + c0 − c2 x + 42c7 + c5 + c1 − c3 x + · · · . 24 2 24 2 Then 30c6 + c4 + 1 1 c0 − c2 = 0 24 2 and 42c7 + c5 + 1 1 c1 − c3 = 0, 24 2 which gives c6 = −c0 /80 and c7 = −19c1 /5040. Thus 1 1 1 y1 (x) = 1 − x2 + x4 − x6 + · · · 2 12 80 and 1 19 7 1 x + ··· . y2 (x) = x − x3 + x5 − 6 30 5040 (b) From part (a) the general solution of the differential equation is y = c1 y1 + c2 y2 . Then y(0) = c1 + c2 · 0 = c1 and y ′ (0) = c1 · 0 + c2 = c2 , so the solution of the initial-value problem is 1 1 1 1 1 19 7 y = y1 + y2 = 1 + x − x2 − x3 + x4 + x5 − x6 − x + ··· . 2 6 12 30 80 5040 272 5.2 (c) y y y 4 4 4 2 2 2 -6 -4 -2 2 4 6 x -6 -4 -2 2 4 6 x -6 -4 -2 -2 -2 -2 -4 -4 -4 y y 4 4 2 2 2 2 4 6 x -6 -4 -2 2 4 6 x 2 4 6 x y 4 -6 -4 -2 (d) Solutions About Singular Points 2 4 6 x -6 -4 -2 -2 -2 -2 -4 -4 -4 y 6 4 2 -6 -4 -2 -2 2 4 6 x -4 -6 EXERCISES 5.2 Solutions About Singular Points 1. Irregular singular point: x = 0 2. Regular singular points: x = 0, −3 3. Irregular singular point: x = 3; regular singular point: x = −3 4. Irregular singular point: x = 1; regular singular point: x = 0 5. Regular singular points: x = 0, ±2i 6. Irregular singular point: x = 5; regular singular point: x = 0 273 5.2 Solutions About Singular Points 7. Regular singular points: x = −3, 2 8. Regular singular points: x = 0, ±i 9. Irregular singular point: x = 0; regular singular points: x = 2, ±5 10. Irregular singular point: x = −1; regular singular points: x = 0, 3 11. Writing the differential equation in the form y ′′ + x 5 y′ + y=0 x−1 x+1 we see that x0 = 1 and x0 = −1 are regular singular points. For x0 = 1 the differential equation can be put in the form (x − 1)2 y ′′ + 5(x − 1)y ′ + x(x − 1)2 y = 0. x+1 In this case p(x) = 5 and q(x) = x(x − 1)2 /(x + 1). For x0 = −1 the differential equation can be put in the form (x + 1)2 y ′′ + 5(x + 1) x+1 ′ y + x(x + 1)y = 0. x−1 In this case p(x) = (x + 1)/(x − 1) and q(x) = x(x + 1). 12. Writing the differential equation in the form y ′′ + x+3 ′ y + 7xy = 0 x we see that x0 = 0 is a regular singular point. Multiplying by x2 , the differential equation can be put in the form x2 y ′′ + x(x + 3)y ′ + 7x3 y = 0. We identify p(x) = x + 3 and q(x) = 7x3 . 13. We identify P (x) = 5/3x + 1 and Q(x) = −1/3x2 , so that p(x) = xP (x) = Then a0 = 35 , b0 = − 13 , and the indicial equation is 5 3 + x and q(x) = x2 Q(x) = − 13 . 5 1 2 1 1 1 r(r − 1) + r − = r2 + r − = (3r2 + 2r − 1) = (3r − 1)(r + 1) = 0. 3 3 3 3 3 3 The indicial roots are 1 3 and −1. Since these do not differ by an integer we expect to find two series solutions using the method of Frobenius. 14. We identify P (x) = 1/x and Q(x) = 10/x, so that p(x) = xP (x) = 1 and q(x) = x2 Q(x) = 10x. Then a0 = 1, b0 = 0, and the indicial equation is r(r − 1) + r = r2 = 0. The indicial roots are 0 and 0. Since these are equal, we expect the method of Frobenius to yield a single series solution. 15. Substituting y = "∞ n=0 cn xn+r into the differential equation and collecting terms, we obtain ∞ & , 2xy ′′ − y ′ + 2y = 2r2 − 3r c0 xr−1 + [2(k + r − 1)(k + r)ck − (k + r)ck + 2ck−1 ]xk+r−1 = 0, k=1 which implies 2r2 − 3r = r(2r − 3) = 0 and (k + r)(2k + 2r − 3)ck + 2ck−1 = 0. 274 5.2 Solutions About Singular Points The indicial roots are r = 0 and r = 3/2. For r = 0 the recurrence relation is ck = − 2ck−1 , k(2k − 3) k = 1, 2, 3, . . . , and c1 = 2c0 , c2 = −2c0 , c3 = 4 c0 , 9 and so on. For r = 3/2 the recurrence relation is ck = − 2ck−1 , (2k + 3)k k = 1, 2, 3, . . . , and 2 2 c1 = − c0 , c2 = c0 , c3 5 35 and so on. The general solution on (0, ∞) is % $ $ 4 3 3/2 2 1− y = C1 1 + 2x − 2x + x + · · · + C2 x 9 16. Substituting y = =− 4 c0 , 945 % 2 2 4 3 2 x+ x − x + ··· . 5 35 945 "∞ cn xn+r into the differential equation and collecting terms, we obtain , , 2xy ′′ + 5y ′ + xy = 2r2 + 3r c0 xr−1 + 2r2 + 7r + 5 c1 xr n=0 + ∞ & k=2 [2(k + r)(k + r − 1)ck + 5(k + r)ck + ck−2 ]xk+r−1 = 0, which implies 2r2 + 3r = r(2r + 3) = 0, , 2 2r + 7r + 5 c1 = 0, and (k + r)(2k + 2r + 3)ck + ck−2 = 0. The indicial roots are r = −3/2 and r = 0, so c1 = 0 . For r = −3/2 the recurrence relation is ck = − ck−2 , (2k − 3)k k = 2, 3, 4, . . . , and 1 c2 = − c0 , 2 and so on. For r = 0 the recurrence relation is ck = − c3 = 0, ck−2 , k(2k + 3) and c4 = 1 c0 , 40 k = 2, 3, 4, . . . , 1 1 c0 , c3 = 0, c4 = c0 , 14 616 and so on. The general solution on (0, ∞) is $ % % $ 1 1 1 4 1 y = C1 x−3/2 1 − x2 + x4 + · · · + C2 1 − x2 + x + ··· . 2 40 14 616 c2 = − 275 5.2 Solutions About Singular Points "∞ cn xn+r into the differential equation and collecting terms, we obtain $ % 0 ∞ / & 7 1 1 ′ 2 r−1 ′′ + 4(k + r)(k + r − 1)ck + (k + r)ck + ck−1 xk+r−1 4xy + y + y = 4r − r c0 x 2 2 2 17. Substituting y = n=0 k=1 = 0, which implies $ % 7 7 4r − r = r 4r − =0 2 2 2 and 1 (k + r)(8k + 8r − 7)ck + ck−1 = 0. 2 The indicial roots are r = 0 and r = 7/8. For r = 0 the recurrence relation is ck = − 2ck−1 , k(8k − 7) and c1 = −2c0 , c2 = k = 1, 2, 3, . . . , 2 c0 , 9 c3 = − 4 c0 , 459 and so on. For r = 7/8 the recurrence relation is ck = − and c1 = − 2ck−1 , (8k + 7)k 2 c0 , 15 c2 = k = 1, 2, 3, . . . , 2 c0 , 345 c3 = − 4 c0 , 32,085 and so on. The general solution on (0, ∞) is % % $ $ 4 3 2 2 4 2 2 x + · · · + C2 x7/8 1 − x + x − x3 + · · · . y = C1 1 − 2x + x2 − 9 459 15 345 32,085 "∞ 18. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain , , , 2x2 y ′′ − xy ′ + x2 + 1 y = 2r2 − 3r + 1 c0 xr + 2r2 + r c1 xr+1 + ∞ & k=2 [2(k + r)(k + r − 1)ck − (k + r)ck + ck + ck−2 ]xk+r = 0, which implies and 2r2 − 3r + 1 = (2r − 1)(r − 1) = 0, , 2 2r + r c1 = 0, [(k + r)(2k + 2r − 3) + 1]ck + ck−2 = 0. The indicial roots are r = 1/2 and r = 1, so c1 = 0. For r = 1/2 the recurrence relation is ck = − and ck−2 , k(2k − 1) 1 c2 = − c0 , 6 k = 2, 3, 4, . . . , c3 = 0, 276 c4 = 1 c0 , 168 5.2 Solutions About Singular Points and so on. For r = 1 the recurrence relation is ck = − ck−2 , k(2k + 1) k = 2, 3, 4, . . . , and 1 1 c0 , c3 = 0, c4 = c0 , 10 360 and so on. The general solution on (0, ∞) is $ % $ % 1 2 1 4 1 2 1 4 1/2 y = C1 x 1− x + x + · · · + C2 x 1 − x + x + ··· . 6 168 10 360 "∞ 19. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain , 3xy ′′ + (2 − x)y ′ − y = 3r2 − r c0 xr−1 c2 = − + ∞ & k=1 [3(k + r − 1)(k + r)ck + 2(k + r)ck − (k + r)ck−1 ]xk+r−1 = 0, which implies 3r2 − r = r(3r − 1) = 0 and (k + r)(3k + 3r − 1)ck − (k + r)ck−1 = 0. The indicial roots are r = 0 and r = 1/3. For r = 0 the recurrence relation is ck−1 ck = , k = 1, 2, 3, . . . , 3k − 1 and 1 1 1 c2 = c3 = c0 , c0 , c0 , 2 10 80 and so on. For r = 1/3 the recurrence relation is ck−1 ck = , k = 1, 2, 3, . . . , 3k and 1 1 1 c1 = c0 , c2 = c0 , c3 = c0 , 3 18 162 and so on. The general solution on (0, ∞) is $ % % $ 1 1 1 3 1 1 1 y = C1 1 + x + x2 + x3 + · · · + C2 x1/3 1 + x + x2 + x + ··· . 2 10 80 3 18 162 "∞ 20. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain $ 0 % $ % ∞ / & 2 2 2 2 ′′ 2 r x y − x− y = r −r+ c0 x + (k + r)(k + r − 1)ck + ck − ck−1 xk+r 9 9 9 c1 = k=1 = 0, which implies 2 r −r+ = 9 2 and $ 2 r− 3 %$ 1 r− 3 % =0 0 / 2 ck − ck−1 = 0. (k + r)(k + r − 1) + 9 277 5.2 Solutions About Singular Points The indicial roots are r = 2/3 and r = 1/3. For r = 2/3 the recurrence relation is ck = 3ck−1 , 3k 2 + k 3 c0 , 4 c2 = k = 1, 2, 3, . . . , and c1 = 9 c0 , 56 c3 = 9 c0 , 560 and so on. For r = 1/3 the recurrence relation is ck = 3ck−1 , 3k 2 − k 3 c0 , 2 c2 = k = 1, 2, 3, . . . , and c1 = 9 c0 , 20 c3 = 9 c0 , 160 and so on. The general solution on (0, ∞) is % % $ $ 9 9 3 9 9 3 3 3 y = C1 x2/3 1 + x + x2 + x + · · · + C2 x1/3 1 + x + x2 + x + ··· . 4 56 560 2 20 160 21. Substituting y = "∞ n=0 cn xn+r into the differential equation and collecting terms, we obtain ∞ & , 2xy ′′ − (3 + 2x)y ′ + y = 2r2 − 5r c0 xr−1 + [2(k + r)(k + r − 1)ck k=1 − 3(k + r)ck − 2(k + r − 1)ck−1 + ck−1 ]xk+r−1 = 0, which implies 2r2 − 5r = r(2r − 5) = 0 and (k + r)(2k + 2r − 5)ck − (2k + 2r − 3)ck−1 = 0. The indicial roots are r = 0 and r = 5/2. For r = 0 the recurrence relation is ck = (2k − 3)ck−1 , k(2k − 5) k = 1, 2, 3, . . . , and c1 = 1 c0 , 3 1 c2 = − c0 , 6 1 c3 = − c0 , 6 and so on. For r = 5/2 the recurrence relation is ck = 2(k + 1)ck−1 , k(2k + 5) k = 1, 2, 3, . . . , and c1 = 4 c0 , 7 c2 = 4 c0 , 21 c3 = 32 c0 , 693 and so on. The general solution on (0, ∞) is $ % % $ 1 1 1 4 32 3 4 y = C1 1 + x − x2 − x3 + · · · + C2 x5/2 1 + x + x2 + x + ··· . 3 6 6 7 21 693 278 5.2 Solutions About Singular Points "∞ cn xn+r into the differential equation and collecting terms, we obtain % $ % $ % $ 4 5 4 y = r2 − c0 xr + r2 + 2r + c1 xr+1 x2 y ′′ + xy ′ + x2 − 9 9 9 0 ∞ / & 4 + (k + r)(k + r − 1)ck + (k + r)ck − ck + ck−2 xk+r 9 22. Substituting y = n=0 k=2 = 0, which implies r2 − and $ %$ % 2 2 4 = r+ r− = 0, 9 3 3 % $ 5 c1 = 0, r2 + 2r + 9 / 0 4 (k + r)2 − ck + ck−2 = 0. 9 The indicial roots are r = −2/3 and r = 2/3, so c1 = 0. For r = −2/3 the recurrence relation is ck = − and 9ck−2 , 3k(3k − 4) 3 c2 = − c0 , 4 and so on. For r = 2/3 the recurrence relation is ck = − and c2 = − c3 = 0, 9ck−2 , 3k(3k + 4) 3 c0 , 20 c3 = 0, k = 2, 3, 4, . . . , c4 = 9 c0 , 128 k = 2, 3, 4, . . . , c4 = 9 c0 , 1,280 and so on. The general solution on (0, ∞) is % % $ $ 9 4 9 3 2 3 2 2/3 4 −2/3 y = C1 x x + · · · + C2 x x + ··· . 1− x + 1− x + 4 128 20 1,280 "∞ 23. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain ∞ & , 9x2 y ′′ + 9x2 y ′ + 2y = 9r2 − 9r + 2 c0 xr + [9(k + r)(k + r − 1)ck + 2ck + 9(k + r − 1)ck−1 ]xk+r = 0, k=1 which implies 9r2 − 9r + 2 = (3r − 1)(3r − 2) = 0 and [9(k + r)(k + r − 1) + 2]ck + 9(k + r − 1)ck−1 = 0. The indicial roots are r = 1/3 and r = 2/3. For r = 1/3 the recurrence relation is ck = − and (3k − 2)ck−1 , k(3k − 1) 1 c1 = − c0 , 2 c2 = 1 c0 , 5 279 k = 1, 2, 3, . . . , c3 = − 7 c0 , 120 5.2 Solutions About Singular Points and so on. For r = 2/3 the recurrence relation is ck = − (3k − 1)ck−1 , k(3k + 1) k = 1, 2, 3, . . . , and 5 1 1 c2 = c0 , c3 = − c0 , c1 = − c0 , 2 28 21 and so on. The general solution on (0, ∞) is $ % % $ 1 1 2 7 3 5 2 1 3 1 1/3 2/3 y = C1 x 1− x+ x − x + · · · + C2 x 1 − x + x − x + ··· . 2 5 120 2 28 21 "∞ 24. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain ∞ & , 2x2 y ′′ + 3xy ′ + (2x − 1)y = 2r2 + r − 1 c0 xr + [2(k + r)(k + r − 1)ck + 3(k + r)ck − ck + 2ck−1 ]xk+r = 0, k=1 which implies 2r2 + r − 1 = (2r − 1)(r + 1) = 0 and [(k + r)(2k + 2r + 1) − 1]ck + 2ck−1 = 0. The indicial roots are r = −1 and r = 1/2. For r = −1 the recurrence relation is ck = − 2ck−1 , k(2k − 3) k = 1, 2, 3, . . . , and c1 = 2c0 , c2 = −2c0 , c3 = 4 c0 , 9 and so on. For r = 1/2 the recurrence relation is ck = − 2ck−1 , k(2k + 3) k = 1, 2, 3, . . . , and 2 4 2 c2 = c0 , c3 = − c0 , c1 = − c0 , 5 35 945 and so on. The general solution on (0, ∞) is $ % % $ 4 2 4 3 2 y = C1 x−1 1 + 2x − 2x2 + x3 + · · · + C2 x1/2 1 − x + x2 − x + ··· . 9 5 35 945 "∞ 25. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain ′′ ′ , 2 - r−1 xy + 2y − xy = r + r c0 x which implies and ∞ & r + r + 3r + 2 c1 x + [(k + r)(k + r − 1)ck + 2(k + r)ck − ck−2 ]xk+r−1 = 0, , 2 k=2 r2 + r = r(r + 1) = 0, , 2 r + 3r + 2 c1 = 0, (k + r)(k + r + 1)ck − ck−2 = 0. 280 5.2 Solutions About Singular Points The indicial roots are r1 = 0 and r2 = −1, so c1 = 0. For r1 = 0 the recurrence relation is ck−2 ck = , k = 2, 3, 4, . . . , k(k + 1) and 1 c0 3! c3 = c5 = c7 = · · · = 0 1 c4 = c0 5! 1 c0 . c2n = (2n + 1)! c2 = For r2 = −1 the recurrence relation is ck = and ck−2 , k(k − 1) k = 2, 3, 4, . . . , 1 c0 2! c3 = c5 = c7 = · · · = 0 1 c4 = c0 4! 1 c2n = c0 . (2n)! c2 = The general solution on (0, ∞) is y = C1 ∞ & 1 1 x2n + C2 x−1 x2n (2n + 1)! (2n)! n=0 n=0 ∞ & 8 7 ∞ ∞ & & 1 1 1 2n+1 2n + C2 C1 x x = x (2n + 1)! (2n)! n=0 n=0 = 1 [C1 sinh x + C2 cosh x]. x "∞ cn xn+r into the differential equation and collecting terms, we obtain $ % $ % $ % 1 1 3 2 ′′ ′ 2 2 r 2 x y + xy + x − y= r − c0 x + r + 2r + c1 xr+1 4 4 4 0 ∞ / & 1 (k + r)(k + r − 1)ck + (k + r)ck − ck + ck−2 xk+r + 4 26. Substituting y = n=0 k=2 = 0, which implies r2 − and $ %$ % 1 1 1 = r− r+ = 0, 4 2 2 $ % 3 r2 + 2r + c1 = 0, 4 / 0 1 (k + r)2 − ck + ck−2 = 0. 4 281 5.2 Solutions About Singular Points The indicial roots are r1 = 1/2 and r2 = −1/2, so c1 = 0. For r1 = 1/2 the recurrence relation is ck = − ck−2 , k(k + 1) k = 2, 3, 4, . . . , and 1 c0 3! c3 = c5 = c7 = · · · = 0 1 c4 = c0 5! (−1)n c2n = c0 . (2n + 1)! c2 = − For r2 = −1/2 the recurrence relation is ck = − ck−2 , k(k − 1) k = 2, 3, 4, . . . , and 1 c0 2! c3 = c5 = c7 = · · · = 0 1 c4 = c0 4! (−1)n c0 . c2n = (2n)! c2 = − The general solution on (0, ∞) is y = C1 x1/2 ∞ ∞ & & (−1)n 2n (−1)n 2n x + C2 x−1/2 x (2n + 1)! (2n)! n=0 n=0 = C1 x−1/2 ∞ ∞ & & (−1)n 2n (−1)n 2n+1 + C2 x−1/2 x x (2n + 1)! (2n)! n=0 n=0 = x−1/2 [C1 sin x + C2 cos x]. 27. Substituting y = "∞ n=0 cn xn+r into the differential equation and collecting terms, we obtain ∞ & , xy ′′ − xy ′ + y = r2 − r c0 xr−1 + [(k + r + 1)(k + r)ck+1 − (k + r)ck + ck ]xk+r = 0 k=0 which implies r2 − r = r(r − 1) = 0 and (k + r + 1)(k + r)ck+1 − (k + r − 1)ck = 0. The indicial roots are r1 = 1 and r2 = 0. For r1 = 1 the recurrence relation is ck+1 = kck , (k + 2)(k + 1) 282 k = 0, 1, 2, . . . , 5.2 Solutions About Singular Points and one solution is y1 = c0 x. A second solution is 4 % $ 3 − −1 dx 3 x 3 e e 1 1 3 1 2 y2 = x dx = x dx = x 1 + x + x + x + · · · dx x2 x2 x2 2 3! % / 0 3 $ 1 1 1 1 2 1 1 1 2 1 3 1 + + + x + x + · · · dx = x − + ln x + x + x + x + · · · =x x2 x 2 3! 4! x 2 12 72 1 1 1 = x ln x − 1 + x2 + x3 + x4 + · · · . 2 12 72 The general solution on (0, ∞) is y = C1 x + C2 y2 (x). "∞ 28. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain , , 3 y ′′ + y ′ − 2y = r2 + 2r c0 xr−2 + r2 + 4r + 3 c1 xr−1 x + ∞ & k=2 [(k + r)(k + r − 1)ck + 3(k + r)ck − 2ck−2 ]xk+r−2 = 0, which implies r2 + 2r = r(r + 2) = 0 , 2 r + 4r + 3 c1 = 0 (k + r)(k + r + 2)ck − 2ck−2 = 0. The indicial roots are r1 = 0 and r2 = −2, so c1 = 0. For r1 = 0 the recurrence relation is 2ck−2 ck = , k = 2, 3, 4, . . . , k(k + 2) and 1 c2 = c0 4 c3 = c5 = c7 = · · · = 0 1 c4 = c0 48 1 c6 = c0 . 1,152 The result is % $ 1 4 1 1 2 y1 = c0 1 + x + x + x6 + · · · . 4 48 1,152 A second solution is 4 3 3 − (3/x)dx dx e dx = y y2 = y 1 , -2 1 1 2 1 4 3 y12 x 1 + 4 x + 48 x + ··· $ % 3 3 1 2 7 4 19 6 1 dx , 1 − x + x + x + · · · dx = y = y1 1 5 4 7 x3 2 48 576 x + 576 x6 + · · · x3 1 + 21 x2 + 48 % 0 / 3 $ 1 1 7 19 3 1 7 2 19 4 1 = y1 − + · · · dx = y − − + · · · + x − x ln x + x x − 1 x3 2x 48 576 2x2 2 96 2,304 / 0 1 1 7 19 4 = − y1 ln x + y − 2 + x2 − x + ··· . 2 2x 96 2,304 283 5.2 Solutions About Singular Points The general solution on (0, ∞) is y = C1 y1 (x) + C2 y2 (x). 29. Substituting y = "∞ n=0 ′′ cn xn+r into the differential equation and collecting terms, we obtain 2 ′ r−1 xy + (1 − x)y − y = r c0 x ∞ & + k=1 [(k + r)(k + r − 1)ck + (k + r)ck − (k + r)ck−1 ]xk+r−1 = 0, which implies r2 = 0 and (k + r)2 ck − (k + r)ck−1 = 0. The indicial roots are r1 = r2 = 0 and the recurrence relation is ck = ck−1 , k k = 1, 2, 3, . . . . One solution is y1 = c0 $ % 1 3 1 2 1 + x + x + x + · · · = c0 ex . 2 3! A second solution is 3 4 3 ex /x 1 −x x dx = e e dx 2x 2x e e x $ % % 3 $ 3 1 2 1 1 3 1 1 2 1 x x 1 − x + x − x + · · · dx = e − 1 + x − x + · · · dx =e x 2 3! x 2 3! 0 / ∞ & 1 3 1 2 (−1)n+1 n x − x + · · · = ex ln x − ex x . = ex ln x − x + 2·2 3 · 3! n · n! n=1 y2 = y 1 e− (1/x−1)dx dx = ex 3 The general solution on (0, ∞) is y = C1 ex + C2 ex 30. Substituting y = "∞ n=0 5 ∞ & (−1)n+1 n ln x − x n · n! n=1 6 . cn xn+r into the differential equation and collecting terms, we obtain xy ′′ + y ′ + y = r2 c0 xr−1 + ∞ & k=1 [(k + r)(k + r − 1)ck + (k + r)ck + ck−1 ]xk+r−1 = 0 which implies r2 = 0 and (k + r)2 ck + ck−1 = 0. The indicial roots are r1 = r2 = 0 and the recurrence relation is ck = − ck−1 , k2 k = 1, 2, 3, . . . . One solution is % $ ∞ & (−1)n n 1 3 1 4 1 y1 = c0 1 − x + 2 x2 − x + x − · · · = c x . 0 2 2 2 (3!) (4!) (n!)2 n=0 284 5.2 Solutions About Singular Points A second solution is y2 = y 1 3 e− 4 (1/x)dx y12 dx = y1 3 3 dx , x 1−x+ 1 2 4x − 1 3 36 x dx , 35 4 x 1 − 2x + 32 x2 − 95 x3 + 288 x − ··· $ % 3 1 5 2 23 3 677 4 = y1 1 + 2x + x + x + x + · · · dx x 2 9 288 % 3 $ 5 23 677 3 1 + 2 + x + x2 + x + · · · dx = y1 x 2 9 288 / 0 5 23 677 4 = y1 ln x + 2x + x2 + x3 + x + ··· 4 27 1,152 $ % 5 23 677 4 = y1 ln x + y1 2x + x2 + x3 + x + ··· . 4 27 1,152 = y1 -2 + ··· The general solution on (0, ∞) is y = C1 y1 (x) + C2 y2 (x). 31. Substituting y = "∞ n+r n=0 cn x into the differential equation and collecting terms, we obtain xy ′′ + (x − 6)y ′ − 3y = (r2 − 7r)c0 xr−1 + which implies ∞ & + (k + r)(k + r − 1)ck + (k + r − 1)ck−1 k=1 . − 6(k + r)ck − 3ck−1 xk+r−1 = 0, r2 − 7r = r(r − 7) = 0 and (k + r)(k + r − 7)ck + (k + r − 4)ck−1 = 0. The indicial roots are r1 = 7 and r2 = 0. For r1 = 7 the recurrence relation is (k + 7)kck + (k + 3)ck−1 = 0, k = 1, 2, 3, . . . , or ck = − Taking c0 ̸= 0 we obtain k+3 ck−1 , k(k + 7) k = 1, 2, 3, . . . . 1 c1 = − c0 2 5 c2 = c0 18 1 c3 = − c0 , 6 and so on. Thus, the indicial root r1 = 7 yields a single solution. Now, for r2 = 0 the recurrence relation is k(k − 7)ck + (k − 4)ck−1 = 0, 285 k = 1, 2, 3, . . . . 5.2 Solutions About Singular Points Then −6c1 − 3c0 = 0 −10c2 − 2c1 = 0 −12c3 − c2 = 0 −12c4 + 0c3 = 0 =⇒ c4 = 0 −10c5 + c4 = 0 =⇒ c5 = 0 −6c6 + 2c5 = 0 =⇒ c6 = 0 0c7 + 3c6 = 0 =⇒ c7 is arbitrary and ck = − Taking c0 ̸= 0 and c7 = 0 we obtain k−4 ck−1 , k(k − 7) k = 8, 9, 10, . . . . 1 c1 = − c0 2 1 c0 c2 = 10 1 c3 = − c0 120 c4 = c5 = c6 = · · · = 0. Taking c0 = 0 and c7 ̸= 0 we obtain c1 = c2 = c3 = c4 = c5 = c6 = 0 1 c8 = − c7 2 5 c9 = c7 36 1 c10 = − c7 , 36 and so on. In this case we obtain the two solutions 1 1 1 3 1 5 1 y1 = 1 − x + x2 − x and y2 = x7 − x8 + x9 − x10 + · · · . 2 10 120 2 36 36 "∞ 32. Substituting y = n=0 cn xn+r into the differential equation and collecting terms, we obtain x(x − 1)y ′′ + 3y ′ − 2y ∞ & , = 4r − r2 c0 xr−1 + [(k + r − 1)(k + r − 12)ck−1 − (k + r)(k + r − 1)ck + 3(k + r)ck − 2ck−1 ]xk+r−1 k=1 = 0, which implies 4r − r2 = r(4 − r) = 0 and −(k + r)(k + r − 4)ck + [(k + r − 1)(k + r − 2) − 2]ck−1 = 0. The indicial roots are r1 = 4 and r2 = 0. For r1 = 4 the recurrence relation is −(k + 4)kck + [(k + 3)(k + 2) − 2]ck−1 = 0 286 5.2 or k+1 ck−1 , k ck = Taking c0 ̸= 0 we obtain Solutions About Singular Points k = 1, 2, 3, . . . . c1 = 2c0 c2 = 3c0 c3 = 4c0 , and so on. Thus, the indicial root r1 = 4 yields a single solution. For r2 = 0 the recurrence relation is or −k(k − 4)ck + k(k − 3)ck−1 = 0, k = 1, 2, 3, . . . , −(k − 4)ck + (k − 3)ck−1 = 0, k = 1, 2, 3, . . . . Then 3c1 − 2c0 = 0 2c2 − c1 = 0 c3 + 0c2 = 0 ⇒ c3 = 0 0c4 + c3 = 0 ⇒ c4 is arbitrary and ck = Taking c0 ̸= 0 and c4 = 0 we obtain (k − 3)ck−1 , k−4 k = 5, 6, 7, . . . . 2 c0 3 1 c2 = c0 3 c3 = c4 = c5 = · · · = 0. c1 = Taking c0 = 0 and c4 ̸= 0 we obtain c1 = c2 = c3 = 0 c5 = 2c4 c6 = 3c4 c7 = 4c4 , and so on. In this case we obtain the two solutions 2 1 y1 = 1 + x + x2 3 3 y2 = x4 + 2x5 + 3x6 + 4x7 + · · · . and 33. (a) From t = 1/x we have dt/dx = −1/x2 = −t2 . Then dy dt dy dy = = −t2 dx dt dx dt and d2 y d = 2 dx dx Now x4 $ dy dx % = d dx $ −t2 1 d2 y + λy = 4 dx2 t $ dy dt t4 % = −t2 d2 y dt dy − 2 dt dx dt dy d2 y + 2t3 dt2 dt % 287 + λy = $ 2t dt dx % = t4 d2 y dy + 2t3 . 2 dt dt d2 y 2 dy + + λy = 0 dt2 t dt 5.2 Solutions About Singular Points becomes t (b) Substituting y = t "∞ n=0 cn t n+r dy d2 y +2 + λty = 0. 2 dt dt into the differential equation and collecting terms, we obtain dy d2 y +2 + λty = (r2 + r)c0 tr−1 + (r2 + 3r + 2)c1 tr 2 dt dt + ∞ & k=2 [(k + r)(k + r − 1)ck + 2(k + r)ck + λck−2 ]tk+r−1 = 0, which implies r2 + r = r(r + 1) = 0, , 2 r + 3r + 2 c1 = 0, and (k + r)(k + r + 1)ck + λck−2 = 0. The indicial roots are r1 = 0 and r2 = −1, so c1 = 0. For r1 = 0 the recurrence relation is ck = − λck−2 , k(k + 1) k = 2, 3, 4, . . . , and λ c0 3! c3 = c5 = c7 = · · · = 0 c2 = − c4 = λ2 c0 5! .. . c2n = (−1)n λn c0 . (2n + 1)! For r2 = −1 the recurrence relation is ck = − λck−2 , k(k − 1) k = 2, 3, 4, . . . , and λ c0 2! c3 = c5 = c7 = · · · = 0 c2 = − c4 = λ2 c0 4! .. . c2n = (−1)n λn c0 . (2n)! 288 5.2 Solutions About Singular Points The general solution on (0, ∞) is y(t) = c1 ∞ ∞ & & (−1)n √ (−1)n √ ( λ t)2n + c2 t−1 ( λ t)2n (2n + 1)! (2n)! n=0 n=0 = / & 0 ∞ ∞ & 1 (−1)n √ (−1)n √ ( λ t)2n+1 + C2 ( λ t)2n C1 t (2n + 1)! (2n)! n=0 n=0 = √ √ 1 [C1 sin λ t + C2 cos λ t ]. t (c) Using t = 1/x, the solution of the original equation is √ √ λ λ y(x) = C1 x sin + C2 x cos . x x 34. (a) From the boundary conditions y(a) = 0, y(b) = 0 we find √ √ λ λ + C2 cos =0 C1 sin a a √ √ λ λ C1 sin + C2 cos = 0. b b Since this is a homogeneous system of linear equations, it will have nontrivial solutions for C1 and C2 if √ ! ! λ ! sin ! a ! √ ! ! λ ! sin b √ ! λ! √ √ √ √ ! cos λ λ λ λ a !! = sin cos − cos sin √ ! a b a b λ! ! cos b 5√ √ 6 $ % √ b−a λ λ λ = sin = sin − = 0. a b ab This will be the case if $ % √ b−a λ = nπ ab or √ λ= nπab nπab = , n = 1, 2, . . . , b−a L or, if λn = Pn b 4 n2 π 2 a2 b2 = . 2 L EI √ √ The critical loads are then Pn = n2 π 2 (a/b)2 EI0 /L2 . Using C2 = −C1 sin( λ/a)/ cos( λ/a) we have √ 0 √ √ / sin( λ/a) λ λ √ cos y = C1 x sin − x x cos( λ/a) √ √ √ 0 √ / λ λ λ λ cos − cos sin = C3 x sin x a x a % $ √ 1 1 = C3 x sin λ − , x a and yn (x) = C3 x sin nπab L $ 1 1 − x a % = C3 x sin 2 nπab 1 a nπab 1 a2 − 1 = C4 x sin 1− . La x L x 289 5.2 Solutions About Singular Points (b) When n = 1, b = 11, and a = 1, we have, y for C4 = 1, 2 $ 1 y1 (x) = x sin 1.1π 1 − x % . 1 1 3 5 7 9 11 x 35. Express the differential equation in standard form: y ′′′ + P (x)y ′′ + Q(x)y ′ + R(x)y = 0. Suppose x0 is a singular point of the differential equation. Then we say that x0 is a regular singular point if (x − x0 )P (x), (x − x0 )2 Q(x), and (x − x0 )3 R(x) are analytic at x = x0 . 36. Substituting y = "∞ n+r n=0 cn x into the first differential equation and collecting terms, we obtain x3 y ′′ + y = c0 xr + ∞ & k=1 [ck + (k + r − 1)(k + r − 2)ck−1 ]xk+r = 0. It follows that c0 = 0 and ck = −(k + r − 1)(k + r − 2)ck−1 . The only solution we obtain is y(x) = 0. Substituting y = "∞ n+r n=0 cn x into the second differential equation and collecting terms, we obtain x2 y ′′ + (3x − 1)y ′ + y = −rc0 + ∞ & k=0 [(k + r + 1)2 ck − (k + r + 1)ck+1 ]xk+r = 0, which implies 2 −rc0 = 0 (k + r + 1) ck − (k + r + 1)ck+1 = 0. If c0 = 0, then the solution of the differential equation is y = 0. Thus, we take r = 0, from which we obtain ck+1 = (k + 1)ck , k = 0, 1, 2, . . . . Letting c0 = 1 we get c1 = 2, c2 = 3!, c3 = 4!, and so on. The solution of the differential equation is then "∞ y = n=0 (n + 1)!xn , which converges only at x = 0. 37. We write the differential equation in the form x2 y ′′ + (b/a)xy ′ + (c/a)y = 0 and identify a0 = b/a and b0 = c/a as in (12) in the text. Then the indicial equation is r(r − 1) + c b r+ =0 a a or ar2 + (b − a)r + c = 0, which is also the auxiliary equation of ax2 y ′′ + bxy ′ + cy = 0. 290 5.3 Special Functions EXERCISES 5.3 Special Functions 1. Since ν 2 = 1/9 the general solution is y = c1 J1/3 (x) + c2 J−1/3 (x). 2. Since ν 2 = 1 the general solution is y = c1 J1 (x) + c2 Y1 (x). 3. Since ν 2 = 25/4 the general solution is y = c1 J5/2 (x) + c2 J−5/2 (x). 4. Since ν 2 = 1/16 the general solution is y = c1 J1/4 (x) + c2 J−1/4 (x). 5. Since ν 2 = 0 the general solution is y = c1 J0 (x) + c2 Y0 (x). 6. Since ν 2 = 4 the general solution is y = c1 J2 (x) + c2 Y2 (x). 7. We identify α = 3 and ν = 2. Then the general solution is y = c1 J2 (3x) + c2 Y2 (3x). 8. We identify α = 6 and ν = 1 2 . Then the general solution is y = c1 J1/2 (6x) + c2 J−1/2 (6x). 9. We identify α = 5 and ν = 23 . Then the general solution is y = c1 J2/3 (5x) + c2 J−2/3 (5x). √ √ √ 10. We identify α = 2 and ν = 8. Then the general solution is y = c1 J8 ( 2x) + c2 Y8 ( 2x). 11. If y = x−1/2 v(x) then 1 y ′ = x−1/2 v ′ (x) − x−3/2 v(x), 2 3 y ′′ = x−1/2 v ′′ (x) − x−3/2 v ′ (x) + x−5/2 v(x), 4 and $ % 1 x2 y ′′ + 2xy ′ + α2 x2 y = x3/2 v ′′ (x) + x1/2 v ′ (x) + α2 x3/2 − x−1/2 v(x) = 0. 4 Multiplying by x1/2 we obtain $ 1 x v (x) + xv (x) + α x − 4 2 ′′ ′ 2 2 % v(x) = 0, whose solution is v = c1 J1/2 (αx) + c2 J−1/2 (αx). Then y = c1 x−1/2 J1/2 (αx) + c2 x−1/2 J−1/2 (αx). √ 12. If y = x v(x) then 1 y ′ = x1/2 v ′ (x) + x−1/2 v(x) 2 1 ′′ 1/2 ′′ y = x v (x) + x−1/2 v ′ (x) − x−3/2 v(x) 4 and $ $ % % 1 1 1 x2 y ′′ + α2 x2 − ν 2 + y = x5/2 v ′′ (x) + x3/2 v ′ (x) − x1/2 v(x) + α2 x2 − ν 2 + x1/2 v(x) 4 4 4 = x5/2 v ′′ (x) + x3/2 v ′ (x) + (α2 x5/2 − ν 2 x1/2 )v(x) = 0. Multiplying by x−1/2 we obtain x2 v ′′ (x) + xv ′ (x) + (α2 x2 − ν 2 )v(x) = 0, √ √ whose solution is v(x) = c1 Jν (αx) + c2 Yν (αx). Then y = c1 x Jν (αx) + c2 x Yν (αx). 291 5.3 Special Functions 13. Write the differential equation in the form y ′′ + (2/x)y ′ + (4/x)y = 0. This is the form of (18) in the text with a = − 21 , c = 1 2 , b = 4, and p = 1, so, by (19) in the text, the general solution is y = x−1/2 [c1 J1 (4x1/2 ) + c2 Y1 (4x1/2 )]. 14. Write the differential equation in the form y ′′ + (3/x)y ′ + y = 0. This is the form of (18) in the text with a = −1, c = 1, b = 1, and p = 1, so, by (19) in the text, the general solution is y = x−1 [c1 J1 (x) + c2 Y1 (x)]. 15. Write the differential equation in the form y ′′ − (1/x)y ′ + y = 0. This is the form of (18) in the text with a = 1, c = 1, b = 1, and p = 1, so, by (19) in the text, the general solution is y = x[c1 J1 (x) + c2 Y1 (x)]. 16. Write the differential equation in the form y ′′ − (5/x)y ′ + y = 0. This is the form of (18) in the text with a = 3, c = 1, b = 1, and p = 2, so, by (19) in the text, the general solution is y = x3 [c1 J3 (x) + c2 Y3 (x)]. 17. Write the differential equation in the form y ′′ + (1 − 2/x2 )y = 0. This is the form of (18) in the text with a = c = 1, b = 1, and p = 23 , so, by (19) in the text, the general solution is 1 2 , y = x1/2 [c1 J3/2 (x) + c2 Y3/2 (x)] = x1/2 [C1 J3/2 (x) + C2 J−3/2 (x)]. 18. Write the differential equation in the form y ′′ + (4 + 1/4x2 )y = 0. This is the form of (18) in the text with a = 12 , c = 1, b = 2, and p = 0, so, by (19) in the text, the general solution is y = x1/2 [c1 J0 (2x) + c2 Y0 (2x)]. 19. Write the differential equation in the form y ′′ + (3/x)y ′ + x2 y = 0. This is the form of (18) in the text with a = −1, c = 2, b = 21 , and p = 21 , so, by (19) in the text, the general solution is / $ % %0 $ 1 2 1 2 y = x−1 c1 J1/2 x + c2 Y1/2 x 2 2 or −1 y=x % %0 $ / $ 1 2 1 2 x + C2 J−1/2 x . C1 J1/2 2 2 20. Write the differential equation in the form y ′′ + (1/x)y ′ + ( 19 x4 − 4/x2 )y = 0. This is the form of (18) in the text with a = 0, c = 3, b = 19 , and p = 32 , so, by (19) in the text, the general solution is y = c1 J2/3 $ 1 3 x 9 % + c2 Y2/3 $ 1 3 x 9 % or y = C1 J2/3 $ 1 3 x 9 % + C2 J−2/3 292 $ % 1 3 x . 9 5.3 Special Functions 21. Using the fact that i2 = −1, along with the definition of Jν (x) in (7) in the text, we have Iν (x) = i−ν Jν (ix) = i−ν = = = = ∞ & ∞ & (−1)n n!Γ(1 + ν + n) n=0 $ ix 2 %2n+ν 1 x 22n+ν (−1)n i2n+ν−ν n!Γ(1 + ν + n) 2 n=0 ∞ & 1 x 22n+ν (−1)n (i2 )n n!Γ(1 + ν + n) 2 n=0 ∞ & 1 x 22n+ν (−1)2n n!Γ(1 + ν + n) 2 n=0 ∞ & 1 x 22n+ν 1 , n!Γ(1 + ν + n) 2 n=0 which is a real function. 22. (a) The differential equation has the form of (18) in the text with 1 2 2c − 2 = 2 =⇒ c = 2 1 1 b2 c2 = −β 2 c2 = −1 =⇒ β = and b = i 2 2 1 2 2 2 a − p c = 0 =⇒ p = . 4 1 − 2a = 0 =⇒ a = Then, by (19) in the text, / $ % %0 $ 1 2 1 2 y = x1/2 c1 J1/4 ix + c2 J−1/4 ix . 2 2 In terms of real functions the general solution can be written / $ % %0 $ 1 2 1 2 1/2 y=x C1 I1/4 x + C2 K1/4 x . 2 2 (b) Write the differential equation in the form y ′′ + (1/x)y ′ − 7x2 y = 0. This is the form of (18) in the text with 1 − 2a = 1 =⇒ a = 0 2c − 2 = 2 =⇒ c = 2 1√ 7 b2 c2 = −β 2 c2 = −7 =⇒ β = 2 a2 − p2 c2 = 0 =⇒ p = 0. and b = Then, by (19) in the text, y = c1 J0 $ 1√ 7 ix2 2 % + c2 Y0 $ % 1√ 7 ix2 . 2 In terms of real functions the general solution can be written % % $ $ 1√ 2 1√ 2 y = C1 I0 7x + C2 K0 7x . 2 2 293 1√ 7i 2 5.3 Special Functions 23. The differential equation has the form of (18) in the text with 1 2 2c − 2 = 0 =⇒ c = 1 1 − 2a = 0 =⇒ a = b2 c2 = 1 =⇒ b = 1 1 a2 − p2 c2 = 0 =⇒ p = . 2 Then, by (19) in the text, 7 9 y = x1/2 [c1 J1/2 (x) + c2 J−1/2 (x)] = x1/2 c1 2 sin x + c2 πx 9 8 2 cos x = C1 sin x + C2 cos x. πx 24. Write the differential equation in the form y ′′ + (4/x)y ′ + (1 + 2/x2 )y = 0. This is the form of (18) in the text with 3 1 − 2a = 4 =⇒ a = − 2 2c − 2 = 0 =⇒ c = 1 b2 c2 = 1 =⇒ b = 1 1 a2 − p2 c2 = 2 =⇒ p = . 2 Then, by (19), (23), and (24) in the text, −3/2 y=x −3/2 [c1 J1/2 (x) + c2 J−1/2 (x)] = x 7 9 c1 2 sin x + c2 πx 9 8 1 1 2 cos x = C1 2 sin x + C2 2 cos x. πx x x 1 2 25. Write the differential equation in the form y ′′ + (2/x)y ′ + ( 16 x − 3/4x2 )y = 0. This is the form of (18) in the text with 1 1 − 2a = 2 =⇒ a = − 2 2c − 2 = 2 =⇒ c = 2 1 1 =⇒ b = b2 c2 = 16 8 3 1 2 2 2 a − p c = − =⇒ p = . 4 2 Then, by (19) in the text, % %0 $ / $ 1 2 1 2 −1/2 x + c2 J−1/2 x y=x c1 J1/2 8 8 7 9 9 $ $ % %8 16 16 1 2 1 2 −1/2 c1 =x sin cos x + c2 x πx2 8 πx2 8 % $ % $ 1 2 1 2 −3/2 −3/2 x + C2 x x . cos = C1 x sin 8 8 26. Write the differential equation in the form y ′′ − (1/x)y ′ + (4 + 3/4x2 )y = 0. This is the form of (18) in the text with 1 − 2a = −1 =⇒ a = 1 2c − 2 = 0 =⇒ c = 1 b2 c2 = 4 =⇒ b = 2 1 3 a2 − p2 c2 = =⇒ p = . 4 2 294 5.3 Special Functions Then, by (19) in the text, y = x[c1 J1/2 (2x) + c2 J−1/2 (2x)] 8 7 9 9 2 2 sin 2x + c2 cos 2x = x c1 π2x π2x = C1 x1/2 sin 2x + C2 x1/2 cos 2x. 27. (a) The recurrence relation follows from −νJν (x) + xJν−1 (x) = − =− = ∞ & ∞ 1 x 22n+ν & (−1)n ν (−1)n 1 x 22n+ν−1 +x n!Γ(1 + ν + n) 2 n!Γ(ν + n) 2 n=0 n=0 ∞ ∞ & 1 x 22n+ν & (−1)n (ν + n) 1 x 2 1 x 22n+ν−1 (−1)n ν + ·2 n!Γ(1 + ν + n) 2 n!Γ(1 + ν + n) 2 2 n=0 n=0 ∞ & (−1)n (2n + ν) 1 x 22n+ν = xJν′ (x). n!Γ(1 + ν + n) 2 n=0 (b) The formula in part (a) is a linear first-order differential equation in Jν (x). An integrating factor for this equation is xν , so d ν [x Jν (x)] = xν Jν−1 (x). dx 28. Subtracting the formula in part (a) of Problem 27 from the formula in Example 5 we obtain 0 = 2νJν (x) − xJν+1 (x) − xJν−1 (x) or 2νJν (x) = xJν+1 (x) + xJν−1 (x). 29. Letting ν = 1 in (21) in the text we have d xJ0 (x) = [xJ1 (x)] dx J0′ (x) 30. From (20) we obtain √ 31. Since Γ( 21 ) = π and we obtain so 3 0 x !r=x ! = xJ1 (x). rJ0 (r) dr = rJ1 (r) ! = −J1 (x), and from (21) we obtain r=0 J0′ (x) $ % 1 (2n − 1)! √ Γ 1− +n = π 2 (n − 1)!22n−1 = J−1 (x). Thus J0′ (x) = J−1 (x) = −J1 (x). n = 1, 2, 3, . . . , ∞ 1 x 22n−1/2 1 1 x 2−1/2 & (−1)n (n − 1)!22n−1 x2n−1/2 (−1)n √ = + n!(2n − 1)!22n−1/2 π n!Γ(1 − 21 + n) 2 Γ( 12 ) 2 n=1 n=0 9 9 9 9 ∞ ∞ 1 2 & (−1)n 21/2 x−1/2 2n 2 2 & (−1)n 2n 2 √ x = =√ + + x = cos x. πx πx n=1 (2n)! πx π x n=1 2n(2n − 1)! π J−1/2 (x) = ∞ & 32. (a) By Problem 28, with ν = 1/2, we obtain J1/2 (x) = xJ3/2 (x) + xJ−1/2 (x) so that 9 $ % 2 sin x − cos x ; J3/2 (x) = πx x with ν = −1/2 we obtain −J−1/2 (x) = xJ1/2 (x) + xJ−3/2 (x) so that 9 2 2 1 cos x J−3/2 (x) = − + sin x ; πx x and with ν = 3/2 we obtain 3J3/2 (x) = xJ5/2 (x) + xJ1/2 (x) so that 9 $ % 2 3 sin x 3 cos x − − sin x . J5/2 (x) = πx x2 x 295 5.3 Special Functions (b) y 1 0.5 -0.5 -1 ν = 1/2 5 y 1 0.5 -0.5 -1 y 1 0.5 10 20 x 15 5 10 ν = −1/2 5 -0.5 -1 y 1 0.5 ν = −3/2 20 x 15 5 -0.5 -1 2 s= α and 10 20 x 15 20 x -0.5 -1 9 10 k −αt/2 e , m 8 6 7 9 5 9 dx dx ds dx 2 k 1 α 2 −αt/2 dx k −αt/2 = − e − = = e dt ds dt dt α m 2 ds m 5 9 6 6 % $ %5 9 d dx dx dx α k −αt/2 k −αt/2 + e e − = dt ds 2 m dt ds m 6 6 5 9 5 9 dx α k −αt/2 d2 x ds k −αt/2 + 2 = e − e ds 2 m ds dt m 6 5 9 % $ dx α k −αt/2 d2 x k −αt = + 2 e e . ds 2 m ds m d2 x d = 2 dt dt $ Then d2 x mα d2 x m 2 + ke−αt x = ke−αt 2 + dt ds 2 Multiplying by 22 /α2 m we have 22 k −αt d2 x 2 e + 2 2 α m ds α : or, since s = (2/α) k/m e−αt/2 , 34. Differentiating y = x1/2 w and 15 ν = 5/2 33. Letting we have y 1 0.5 ,2 s2 9 9 k −αt/2 dx e + ke−αt x = 0. m ds k −αt/2 dx 22 k −αt e + 2 e x=0 m ds α m dx d2 x +s + s2 x = 0. ds2 ds - with respect to 23 αx3/2 we obtain $ % % $ 2 3/2 2 3/2 1 y ′ = x1/2 w′ αx1/2 + x−1/2 w αx αx 3 2 3 % % $ $ 2 3/2 2 3/2 1/2 ′ ′′ ′′ αx αx y = αxw αx + αw 3 3 $ $ % % 1 2 3/2 2 3/2 1 −3/2 ′ + αw αx w αx − x . 2 3 4 3 3/2 3 αx Then, after combining terms and simplifying, we have % 0 $ / 1 3 y ′′ + α2 xy = α αx3/2 w′′ + w′ + αx3/2 − w = 0. 2 4αx3/2 296 ν = 3/2 5 10 15 20 x 5.3 Letting t = 32 αx3/2 or αx3/2 = 23 t this differential equation becomes / $ % 0 1 3 α 2 ′′ t w (t) + tw′ (t) + t2 − w(t) = 0, 2 t 9 Special Functions t > 0. 35. (a) By Problem 34, a solution of Airy’s equation is y = x1/2 w( 32 αx3/2 ), where w(t) = c1 J1/3 (t) + c2 J−1/3 (t) is a solution of Bessel’s equation of order 31 . Thus, the general solution of Airy’s equation for x > 0 is $ % % % $ $ 2 3/2 2 3/2 2 3/2 y = x1/2 w = c1 x1/2 J1/3 + c2 x1/2 J−1/3 . αx αx αx 3 3 3 (b) Airy’s equation, y ′′ + α2 xy = 0, has the form of (18) in the text with 1 2 3 2c − 2 = 1 =⇒ c = 2 2 2 2 2 b c = α =⇒ b = α 3 1 2 2 2 a − p c = 0 =⇒ p = . 3 1 − 2a = 0 =⇒ a = Then, by (19) in the text, 1/2 y=x / $ % %0 $ 2 3/2 2 3/2 c1 J1/3 αx αx + c2 J−1/3 . 3 3 36. The general solution of the differential equation is y(x) = c1 J0 (αx) + c2 Y0 (αx). In order to satisfy the conditions that limx→0+ y(x) and limx→0+ y ′ (x) are finite we are forced to define c2 = 0. Thus, y(x) = c1 J0 (αx). The second boundary condition, y(2) = 0, implies c1 = 0 or J0 (2α) = 0. In order to have a nontrivial solution we require that J0 (2α) = 0. From Table 5.1, the first three positive zeros of J0 are found to be 2α1 = 2.4048, 2α2 = 5.5201, 2α3 = 8.6537 and so α1 = 1.2024, α2 = 2.7601, α3 = 4.3269. The eigenfunctions corresponding to the eigenvalues λ1 = α12 , λ2 = α22 , λ3 = α32 are J0 (1.2024x), J0 (2.7601x), and J0 (4.3269x). 37. (a) The differential equation y ′′ + (λ/x)y = 0 has the form of (18) in the text with 1 2 1 2c − 2 = −1 =⇒ c = 2 √ 2 2 b c = λ =⇒ b = 2 λ 1 − 2a = 0 =⇒ a = a2 − p2 c2 = 0 =⇒ p = 1. Then, by (19) in the text, √ √ y = x1/2 [c1 J1 (2 λx ) + c2 Y1 (2 λx )]. (b) We first note that y = J1 (t) is a solution of Bessel’s equation, t2 y ′′ + ty ′ + (t2 − 1)y = 0, with ν = 1. That is, t2 J1′′ (t) + tJ1′ (t) + (t2 − 1)J1 (t) = 0, 297 5.3 Special Functions √ or, letting t = 2 x , Now, if y = √ √ √ √ √ 4xJ1′′ (2 x ) + 2 xJ1′ (2 x ) + (4x − 1)J1 (2 x ) = 0. √ xJ1 (2 x ), we have y′ = and √ √ √ √ √ 1 1 1 x J1′ (2 x ) √ + √ J1 (2 x ) = J1′ (2 x ) + x−1/2 J1 (2 x ) 2 x 2 x √ √ 1 ′ √ 1 y ′′ = x−1/2 J1′′ (2 x ) + J1 (2 x ) − x−3/2 J1 (2 x ). 2x 4 Then √ √ √ √ √ 1 1 x J1′′ 2 x + J1′ (2 x ) − x−1/2 J1 (2 x ) + x J(2 x ) 2 4 √ √ √ √ √ 1 = √ [4xJ1′′ (2 x ) + 2 x J1′ (2 x ) − J1 (2 x ) + 4xJ(2 x )] 4 x xy ′′ + y = √ = 0, √ and y = √ x J1 (2 x ) is a solution of Airy’s differential equation. 38. We see from the graphs below that the graphs of the modified Bessel functions are not oscillatory, while those of the Bessel functions, shown in Figures 5.3 and 5.4 in the text, are oscillatory. I0 I1 I2 20 20 20 15 15 15 10 10 10 5 5 5 1 2 3 4 5 x 1 2 3 5 x 4 K0 5 K1 5 K2 5 4 4 4 3 3 3 2 2 2 1 1 1 1 2 3 4 5 x 1 2 3 4 5 x 39. (a) We identify m = 4, k = 1, and α = 0.1. Then x(t) = c1 J0 (10e−0.05t ) + c2 Y0 (10e−0.05t ) and x′ (t) = −0.5c1 J0′ (10e−0.05t ) − 0.5c2 Y0′ (10e−0.05t ). Now x(0) = 1 and x′ (0) = −1/2 imply c1 J0 (10) + c2 Y0 (10) = 1 c1 J0′ (10) + c2 Y0′ (10) = 1. 298 1 2 3 4 5 x 1 2 3 4 5 x 5.3 Special Functions Using Cramer’s rule we obtain c1 = Y0′ (10) − Y0 (10) J0 (10)Y0′ (10) − J0′ (10)Y0 (10) c2 = J0 (10) − J0′ (10) . J0 (10)Y0′ (10) − J0′ (10)Y0 (10) and Using Y0′ = −Y1 and J0′ = −J1 and Table 5.2 we find c1 = −4.7860 and c2 = −3.1803. Thus x(t) = −4.7860J0 (10e−0.05t ) − 3.1803Y0 (10e−0.05t ). x (b) 10 5 t −5 50 40. (a) Identifying α = 1 2 100 150 200 , the general solution of x′′ + 41 tx = 0 is % % $ $ 1 3/2 1 3/2 1/2 1/2 x x + c2 x J−1/3 . x(t) = c1 x J1/3 3 3 Solving the system x(0.1) = 1, x′ (0.1) = − 21 we find c1 = −0.809264 and c2 = 0.782397. x (b) 1 t −1 50 150 100 200 41. (a) Letting t = L − x, the boundary-value problem becomes d2 θ + α2 tθ = 0, dt2 θ′ (0) = 0, θ(L) = 0, where α2 = δg/EI. This is Airy’s differential equation, so by Problem 35 its solution is $ % % $ 2 3/2 2 3/2 1/2 1/2 y = c1 t J1/3 αt αt + c2 t J−1/3 = c1 θ1 (t) + c2 θ2 (t). 3 3 (b) Looking at the series forms of θ1 and θ2 we see that θ1′ (0) ̸= 0, while θ2′ (0) = 0. Thus, the boundary condition θ′ (0) = 0 implies c1 = 0, and so √ θ(t) = c2 t J−1/3 From θ(L) = 0 we have √ c2 L J−1/3 $ $ % 2 3/2 αt . 3 2 3/2 αL 3 % = 0, so either c2 = 0, in which case θ(t) = 0, or J−1/3 ( 32 αL3/2 ) = 0. The column will just start to bend when L is the length corresponding to the smallest positive zero of J−1/3 . 299 5.3 Special Functions (c) Using Mathematica, the first positive root of J−1/3 (x) is x1 ≈ 1.86635. Thus 23 αL3/2 = 1.86635 implies $ 01/3 %2/3 / 3(1.86635) 9EI L= = (1.86635)2 2α 4δg 01/3 / 9(2.6 × 107 )π(0.05)4 /4 2 ≈ 76.9 in. (1.86635) = 4(0.28)π(0.05)2 42. (a) Writing the differential equation in the form xy ′′ + (P L/M )y = 0, we identify λ = P L/M . Problem 37 the solution of this differential equation is 2 2 1 : 1 : √ √ y = c1 x J1 2 P Lx/M + c2 x Y1 2 P Lx/M . From Now J1 (0) = 0, so y(0) = 0 implies c2 = 0 and 2 1 : √ y = c1 x J1 2 P Lx/M . √ (b) From y(L) = 0 we have y = J1 (2L P M ) = 0. The first positive zero of J1 is 3.8317 so, solving : 2L P1 /M = 3.8317, we find P1 = 3.6705M/L2 . Therefore, 6 5 9 $ % √ √ 3.8317 √ 3.6705x √ y1 (x) = c1 x J1 2 = c1 x J1 x . L L (c) For c1 = 1 and L = 1 the graph of y1 = √ √ x J1 (3.8317 x ) y 0.3 is shown. 0.2 0.1 x 0.2 0.4 0.6 0.8 1 43. (a) Since l′ = v, we integrate to obtain l(t) = vt + c. Now l(0) = l0 implies c = l0 , so l(t) = vt + l0 . Using sin θ ≈ θ in l d2 θ/dt2 + 2l′ dθ/dt + g sin θ = 0 gives (l0 + vt) dθ d2 θ + 2v + gθ = 0. dt2 dt (b) Dividing by v, the differential equation in part (a) becomes l0 + vt d2 θ dθ g +2 + θ = 0. 2 v dt dt v Letting x = (l0 + vt)/v = t + l0 /v we have dx/dt = 1, so dθ dx dθ dθ = = dt dx dt dx and d2 θ d(dθ/dt) d(dθ/dx) dx d2 θ = . = = dt2 dt dx dt dx2 Thus, the differential equation becomes x d2 θ dθ g +2 + θ=0 2 dx dx v or 300 d2 θ 2 dθ g + + θ = 0. 2 dx x dx vx 5.3 Special Functions (c) The differential equation in part (b) has the form of (18) in the text with 1 − 2a = 2 =⇒ a = − 1 2 1 2 9 g g b2 c2 = =⇒ b = 2 v v 2c − 2 = −1 =⇒ c = a2 − p2 c2 = 0 =⇒ p = 1. Then, by (19) in the text, / $ 9 % %0 $ 9 g 1/2 g 1/2 θ(x) = x c1 J1 2 + c2 Y1 2 x x v v $ / $ % %0 9 2: v 2: θ(t) = c1 J1 g(l0 + vt) + c2 Y1 g(l0 + vt) . l0 + vt v v −1/2 or (d) To simplify calculations, let 9 g 1/2 2: u= g(l0 + vt) = 2 x , v v √ and at t = 0 let u0 = 2 gl0 /v. The general solution for θ(t) can then be written θ = C1 u−1 J1 (u) + C2 u−1 Y1 (u). (1) Before applying the initial conditions, note that dθ dθ du = dt du dt so when dθ/dt = 0 at t = 0 we have dθ/du = 0 at u = u0 . Also, d −1 d −1 dθ = C1 [u J1 (u)] + C2 [u Y1 (u)] du du du which, in view of (20) in the text, is the same as dθ = −C1 u−1 J2 (u) − C2 u−1 Y2 (u). du Now at t = 0, or u = u0 , (1) and (2) give the system −1 C1 u−1 0 J1 (u0 ) + C2 u0 Y1 (u0 ) = θ0 −1 C1 u−1 0 J2 (u0 ) + C2 u0 Y2 (u0 ) = 0 whose solution is easily obtained using Cramer’s rule: C1 = u0 θ0 Y2 (u0 ) , J1 (u0 )Y2 (u0 ) − J2 (u0 )Y1 (u0 ) C2 = −u0 θ0 J2 (u0 ) . J1 (u0 )Y2 (u0 ) − J2 (u0 )Y1 (u0 ) In view of the given identity these results simplify to π C1 = − u20 θ0 Y2 (u0 ) 2 The solution is then θ= and C2 = π 2 u θ0 J2 (u0 ). 2 0 / 0 Y1 (u) J1 (u) π 2 u0 θ0 −Y2 (u0 ) + J2 (u0 ) . 2 u u 301 (2) 5.3 Special Functions : √ Returning to u = (2/v) g(l0 + vt) and u0 = (2/v) gl0 , we have $ $ % %⎤ ⎡ 2: 2: √ $ $ % J1 % g(l + vt) g(l + vt) Y 0 0 1 : ⎥ π gl0 θ0 ⎢ 2: v v ⎢−Y2 2 gl0 ⎥. √ √ + J gl θ(t) = 2 0 ⎣ ⎦ v v v l0 + vt l0 + vt 1 radian, and v = 60 ft/s, the above function is √ √ J1 (480 2(1 + t/60)) Y1 (480 2(1 + t/60)) : : θ(t) = −1.69045 − 2.79381 . 1 + t/60 1 + t/60 (e) When l0 = 1 ft, θ0 = 1 10 The plots of θ(t) on [0, 10], [0, 30], and [0, 60] are Θ !t" 0.1 Θ !t" 0.1 0.05 Θ !t" 0.1 0.05 2 4 6 8 10 t 0.05 5 10 15 20 25 30 t 10 -0.05 -0.05 -0.05 -0.1 -0.1 -0.1 (f ) The graphs indicate that θ(t) decreases as l increases. The 20 30 40 50 60 t Θ !t" 0.1 graph of θ(t) on [0, 300] is shown. 0.05 50 100 150 200 -0.05 -0.1 44. (a) From (26) in the text, we have $ % 6·7 2 4·6·7·9 4 2 · 4 · 6 · 7 · 9 · 11 6 P6 (x) = c0 1 − x + x = x , 2! 4! 6! where c0 = (−1)3 Thus, 5 P6 (x) = − 16 $ 5 1·3·5 =− . 2·4·6 16 231 6 x 1 − 21x + 63x − 5 2 4 % = 1 (231x6 − 315x4 + 105x2 − 5). 16 Also, from (26) in the text we have $ % 6 · 9 3 4 · 6 · 9 · 11 5 2 · 4 · 6 · 9 · 11 · 13 7 P7 (x) = c1 x − x + x − x 3! 5! 7! where c1 = (−1)3 1·3·5·7 35 =− . 2·4·6 16 Thus $ % 99 5 429 7 1 35 3 x − 9x + x − x = (429x7 − 693x5 + 315x3 − 35x). P7 (x) = − 16 5 35 16 , , (b) P6 (x) satisfies 1 − x2 y ′′ − 2xy ′ + 42y = 0 and P7 (x) satisfies 1 − x2 y ′′ − 2xy ′ + 56y = 0. 302 250 300 t 5.3 Special Functions 45. The recurrence relation can be written Pk+1 (x) = k = 1: P2 (x) = k = 2: P3 (x) = k = 3: P4 (x) = k = 4: P5 (x) = k = 5: P6 (x) = k = 6: P7 (x) = = k 2k + 1 xPk (x) − Pk−1 (x), k+1 k+1 k = 2, 3, 4, . . . . 3 2 1 x − 2 2 $ % 3 2 1 5 3 2 5 x x − − x = x3 − x 3 2 2 3 2 2 $ % $ % 7 5 3 3 3 3 2 1 35 4 30 2 3 x x − x − x − x − x + = 4 2 2 4 2 2 8 8 8 $ % % $ 35 4 30 2 3 63 5 35 3 15 4 5 3 3 9 x x − x + x − x = x − x + x − 5 8 8 8 5 2 2 8 4 8 $ % $ % 11 5 63 5 35 3 15 5 35 4 30 2 3 231 6 315 4 105 2 x x − x + x − x − x + x − x + x − = 6 8 4 8 6 8 8 8 16 16 16 16 $ % $ % 5 6 63 5 35 3 15 231 6 315 4 105 2 13 x x − x + x − − x − x + x 7 16 16 16 16 7 8 4 8 429 7 693 5 315 3 35 x − x + x − x 16 16 16 16 46. If x = cos θ then dy dy = − sin θ , dθ dx d2 y dy d2 y = sin2 θ 2 − cos θ , 2 dθ dx dx and That is, / 0 , - d2 y dy dy d2 y 2 + n(n + 1)(sin θ)y = sin θ 1 − cos θ − 2 cos θ + n(n + 1)y = 0. sin θ 2 + cos θ dθ dθ dx2 dx , 1 − x2 - d2 y dy − 2x + n(n + 1)y = 0. 2 dx dx 47. The only solutions bounded on [−1, 1] are y = cPn (x), c a constant and n = 0, 1, 2, . . . . By (iv) of the properties of the Legendre polynomials, y(0) = 0 or Pn (0) = 0 implies n must be odd. Thus the first three positive eigenvalues correspond to n = 1, 3, and 5 or λ1 = 1 · 2, λ2 = 3 · 4 = 12, and λ3 = 5 · 6 = 30. We can take the eigenfunctions to be y1 = P1 (x), y2 = P3 (x), and y3 = P5 (x). 48. Using a CAS we find 1 d P1 (x) = (x2 − 1)1 = x 2 dx 1 d2 (x2 − 1)2 = P2 (x) = 2 2 2! dx2 1 d3 P3 (x) = 3 (x2 − 1)3 = 2 3! dx3 1 d4 P4 (x) = 4 (x2 − 1)4 = 2 4! dx4 1 d5 (x2 − 1)5 = P5 (x) = 5 2 5! dx5 1 d6 P6 (x) = 6 (x2 − 1)6 = 2 6! dx6 1 d7 P7 (x) = 7 (x2 − 1)7 = 2 7! dx7 1 (3x2 − 1) 2 1 (5x3 − 3x) 2 1 (35x4 − 30x2 + 3) 8 1 (63x5 − 70x3 + 15x) 8 1 (231x6 − 315x4 + 105x2 − 5) 16 1 (429x7 − 693x5 + 315x3 − 35x) 16 303 5.3 Special Functions P1 1 49. P2 1 0.5 -1 -0.5 P3 1 0.5 0.5 1 x -1 -0.5 P4 1 0.5 0.5 1 x -1 -0.5 0.5 0.5 1 x -1 -0.5 -0.5 -0.5 -0.5 -0.5 -1 -1 -1 -1 P5 1 P6 1 0.5 -1 -0.5 1 x -1 -0.5 1 x P7 1 0.5 0.5 0.5 0.5 0.5 1 x -1 -0.5 -0.5 -0.5 -0.5 -1 -1 -1 0.5 1 x 50. Zeros of Legendre polynomials for n ≥ 1 are P1 (x) : 0 P2 (x) : ±0.57735 P3 (x) : 0, ±0.77460 P4 (x) : ±0.33998, ±0.86115 P5 (x) : 0, ±0.53847, ±0.90618 P6 (x) : ±0.23862, ±0.66121, ±0.93247 P7 (x) : 0, ±0.40585, ±0.74153 , ±0.94911 P10 (x) : ±0.14887, ±0.43340, ±0.67941, ±0.86506, ±0.097391 The zeros of any Legendre polynomial are in the interval (−1, 1) and are symmetric with respect to 0. CHAPTER 5 REVIEW EXERCISES 1. False; J1 (x) and J−1 (x) are not linearly independent when ν is a positive integer. (In this case ν = 1). The general solution of x2 y ′′ + xy ′ + (x2 − 1)y = 0 is y = c1 J1 (x) + c2 Y1 (x). 2. False; y = x is a solution that is analytic at x = 0. 3. x = −1 is the nearest singular point to the ordinary point x = 0. Theorem 5.1 guarantees the existence of two "∞ power series solutions y = n=1 cn xn of the differential equation that converge at least for −1 < x < 1. Since − 12 ≤ x ≤ 21 is properly contained in −1 < x < 1, both power series must converge for all points contained in − 12 ≤ x ≤ 1 2 . 304 CHAPTER 5 REVIEW EXERCISES 4. The easiest way to solve the system 2c2 + 2c1 + c0 = 0 6c3 + 4c2 + c1 = 0 1 12c4 + 6c3 − c1 + c2 = 0 3 2 20c5 + 8c4 − c2 + c3 = 0 3 is to choose, in turn, c0 ̸= 0, c1 = 0 and c0 = 0, c1 ̸= 0. Assuming that c0 = ̸ 0, c1 = 0, we have 1 c2 = − c0 2 1 2 c3 = − c2 = c0 3 3 1 1 1 c4 = − c3 − c2 = − c0 2 12 8 1 1 1 2 c0 ; c5 = − c4 + c2 − c3 = 5 30 20 60 whereas the assumption that c0 = 0, c1 ̸= 0 implies c2 = −c1 2 c3 = − c2 − 3 1 c4 = − c3 + 2 2 c5 = − c4 + 5 1 1 c1 = c1 6 2 1 1 5 c1 − c2 = − c1 36 12 36 1 1 1 c2 − c3 = − c1 . 30 20 360 five terms of two power series solutions are then / 0 1 1 1 1 y1 (x) = c0 1 − x2 + x3 − x4 + x5 + · · · 2 3 8 60 and 0 / 5 1 5 1 x + ··· . y2 (x) = c1 x − x2 + x3 − x4 − 2 36 360 5. The interval of convergence is centered at 4. Since the series converges at −2, it converges at least on the interval [−2, 10). Since it diverges at 13, it converges at most on the interval [−5, 13). Thus, at −7 it does not converge, at 0 and 7 it does converge, and at 10 and 11 it might converge. 6. We have x3 x5 x− + − ··· sin x 2x5 x3 6 120 f (x) = = + + ··· . =x+ 2 4 cos x 3 15 x x 1− + − ··· 2 24 7. The differential equation (x3 − x2 )y ′′ + y ′ + y = 0 has a regular singular point at x = 1 and an irregular singular point at x = 0. 8. The differential equation (x − 1)(x + 3)y ′′ + y = 0 has regular singular points at x = 1 and x = −3. "∞ 9. Substituting y = n=0 cn xn+r into the differential equation we obtain ∞ & , 2xy ′′ + y ′ + y = 2r2 − r c0 xr−1 + [2(k + r)(k + r − 1)ck + (k + r)ck + ck−1 ]xk+r−1 = 0 k=1 305 CHAPTER 5 REVIEW EXERCISES which implies 2r2 − r = r(2r − 1) = 0 and (k + r)(2k + 2r − 1)ck + ck−1 = 0. The indicial roots are r = 0 and r = 1/2. For r = 0 the recurrence relation is ck−1 ck = − , k = 1, 2, 3, . . . , k(2k − 1) so 1 1 c3 = − c0 . c1 = −c0 , c2 = c0 , 6 90 For r = 1/2 the recurrence relation is ck−1 ck = − , k = 1, 2, 3, . . . , k(2k + 1) so 1 1 1 c1 = − c0 , c2 = c0 , c3 = − c0 . 3 30 630 Two linearly independent solutions are 1 1 y1 = 1 − x + x2 − x3 + · · · 6 90 and % $ 1 3 1 1 x + ··· . y2 = x1/2 1 − x + x2 − 3 30 630 "∞ 10. Substituting y = n=0 cn xn into the differential equation we have y ′′ − xy ′ − y = = ∞ & n=2 ' n(n − 1)cn xn−2 − () ∞ & k=0 n=1 ' * k=n−2 ∞ & ncn xn − () k=n (k + 2)(k + 1)ck+2 xk − = 2c2 − c0 + Thus ∞ & k=1 ∞ & k=1 * ∞ & n=0 ' () * 1 c0 2 1 = ck , k+2 c2 = Choosing c0 = 1 and c1 = 0 we find ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 − (k + 1)ck ]xk = 0. (k + 2)(k + 1)ck+2 − (k + 1)ck = 0 ck+2 k=n kck xk − 2c2 − c0 = 0 and cn xn k = 1, 2, 3, . . . . 1 2 c3 = c5 = c7 = · · · = 0 1 c4 = 8 1 c6 = 48 c2 = 306 CHAPTER 5 REVIEW EXERCISES and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 1 c3 = 3 1 c5 = 15 1 c7 = 105 and so on. Thus, two solutions are 1 1 1 y1 = 1 + x2 + x4 + x6 + · · · 2 8 48 and 1 1 1 7 y2 = x + x3 + x5 + x + ···. 3 15 105 11. Substituting y = "∞ n=0 cn xn into the differential equation we obtain (x − 1)y ′′ + 3y = (−2c2 + 3c0 ) + ∞ & k=1 [(k + 1)kck+1 − (k + 2)(k + 1)ck+2 + 3ck ]xk = 0 which implies c2 = 3c0 /2 and ck+2 = (k + 1)kck+1 + 3ck , (k + 2)(k + 1) k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 3 , 2 c3 = 1 , 2 c4 = 5 8 c2 = 0, c3 = 1 , 2 c4 = 1 4 c2 = and so on. For c0 = 0 and c1 = 1 we obtain and so on. Thus, two solutions are 1 5 3 y1 = 1 + x2 + x3 + x4 + · · · 2 2 8 and 1 1 y2 = x + x3 + x4 + · · · . 2 4 12. Substituting y = "∞ n=0 cn xn into the differential equation we obtain y ′′ − x2 y ′ + xy = 2c2 + (6c3 + c0 )x + which implies c2 = 0, c3 = −c0 /6, and ck+3 = Choosing c0 = 1 and c1 = 0 we find ∞ & k=1 [(k + 3)(k + 2)ck+3 − (k − 1)ck ]xk+1 = 0 k−1 ck , (k + 3)(k + 2) k = 1, 2, 3, . . . . 1 6 c4 = c7 = c10 = · · · = 0 c3 = − c5 = c8 = c11 = · · · = 0 1 c6 = − 90 307 CHAPTER 5 REVIEW EXERCISES and so on. For c0 = 0 and c1 = 1 we obtain c3 = c6 = c9 = · · · = 0 c4 = c7 = c10 = · · · = 0 c5 = c8 = c11 = · · · = 0 and so on. Thus, two solutions are 1 1 y1 = 1 − x3 − x6 − · · · 6 90 13. Substituting y = "∞ n+r n=0 cn x and y2 = x. into the differential equation, we obtain xy ′′ − (x + 2)y ′ + 2y = (r2 − 3r)c0 xr−1 + ∞ & k=1 [(k + r)(k + r − 3)ck − (k + r − 3)ck−1 ]xk+r−1 = 0, which implies r2 − 3r = r(r − 3) = 0 and (k + r)(k + r − 3)ck − (k + r − 3)ck−1 = 0. The indicial roots are r1 = 3 and r2 = 0. For r2 = 0 the recurrence relation is k(k − 3)ck − (k − 3)ck−1 = 0, k = 1, 2, 3, . . . . Then c1 − c0 = 0 2c2 − c1 = 0 0c3 − 0c2 = 0 =⇒ c3 is arbitrary and ck = Taking c0 ̸= 0 and c3 = 0 we obtain 1 ck−1 , k k = 4, 5, 6, . . . . c1 = c0 1 c2 = c0 2 c3 = c4 = c5 = · · · = 0. Taking c0 = 0 and c3 ̸= 0 we obtain c0 = c1 = c2 = 0 6 1 c4 = c3 = c3 4 4! 6 1 c3 = c3 c5 = 5·4 5! 6 1 c3 = c3 , c6 = 6·5·4 6! and so on. In this case we obtain the two solutions and 1 y1 = 1 + x + x2 2 y2 = x3 + $ % 6 6 1 6 4 x + x5 + x6 + · · · = 6ex − 6 1 + x + x2 . 4! 5! 6! 2 308 CHAPTER 5 REVIEW EXERCISES 14. Substituting y = "∞ n n=0 cn x into the differential equation we have % ∞ & 1 2 1 4 1 6 ′′ 2 3 4 cn xn (cos x)y + y = 1 − x + x − x + · · · (2c2 + 6c3 x + 12c4 x + 20c5 x + 30c6 x + · · ·) + 2 24 720 n=0 / $ % 0 1 = 2c2 + 6c3 x + (12c4 − c2 )x2 + (20c5 − 3c3 )x3 + 30c6 − 6c4 + c2 x4 + · · · 12 $ + [c0 + c1 x + c2 x2 + c3 x3 + c4 x4 + · · · ] % $ 1 = (c0 + 2c2 ) + (c1 + 6c3 )x + 12c4 x2 + (20c5 − 2c3 )x3 + 30c6 − 5c4 + c2 x4 + · · · 12 = 0. Thus c0 + 2c2 = 0 c1 + 6c3 = 0 12c4 = 0 20c5 − 2c3 = 0 1 30c6 − 5c4 + c2 = 0 12 and 1 c2 = − c0 2 1 c3 = − c1 6 c4 = 0 1 c5 = c3 10 1 1 c2 . c6 = c4 − 6 360 Choosing c0 = 1 and c1 = 0 we find 1 1 c2 = − , c3 = 0, c4 = 0, c5 = 0, c6 = 2 720 and so on. For c0 = 0 and c1 = 1 we find 1 1 c2 = 0, c3 = − , c4 = 0, c5 = − , c6 = 0 6 60 and so on. Thus, two solutions are 1 6 1 y1 = 1 − x2 + x + ··· 2 720 15. Substituting y = "∞ n n=0 cn x 1 1 y2 = x − x3 − x5 + · · · . 6 60 and into the differential equation we have y ′′ + xy ′ + 2y = ∞ & n=2 = ' ∞ & n(n − 1)cn xn−2 + () k=n−2 * ∞ & ncn xn + 2 n=1 ' k=n = 2c2 + 2c0 + ∞ & k=1 ∞ & cn xn n=0 () (k + 2)(k + 1)ck+2 xk + k=0 ∞ & * ' () * k=n kck xk + 2 ∞ & ck xk k=0 [(k + 2)(k + 1)ck+2 + (k + 2)ck ]xk = 0. k=1 309 CHAPTER 5 REVIEW EXERCISES Thus 2c2 + 2c0 = 0 (k + 2)(k + 1)ck+2 + (k + 2)ck = 0 and c2 = −c0 ck+2 = − 1 ck , k+1 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find c2 = −1 c3 = c5 = c7 = · · · = 0 1 c4 = 3 1 c6 = − 15 and so on. For c0 = 0 and c1 = 1 we obtain c2 = c4 = c6 = · · · = 0 1 c3 = − 2 1 c5 = 8 1 c7 = − 48 and so on. Thus, the general solution is $ % % $ 1 1 1 1 1 y = C0 1 − x2 + x4 − x6 + · · · + C1 x − x3 + x5 − x7 + · · · 3 15 2 8 48 and $ % % $ 4 2 5 7 3 y ′ = C0 −2x + x3 − x5 + · · · + C1 1 − x2 + x4 − x6 + · · · . 3 5 2 8 48 Setting y(0) = 3 and y ′ (0) = −2 we find c0 = 3 and c1 = −2. Therefore, the solution of the initial-value problem is 1 1 1 y = 3 − 2x − 3x2 + x3 + x4 − x5 − x6 + x7 + · · · . 4 5 24 "∞ 16. Substituting y = n=0 cn xn into the differential equation we have (x + 2)y ′′ + 3y = ∞ & n=2 = ' ∞ & n(n − 1)cn xn−1 + 2 () k=n−1 * (k + 1)kck+1 xk + 2 k=1 = 4c2 + 3c0 + ∞ & n=2 ' ∞ & n(n − 1)cn xn−2 + 3 () k=n−2 * cn xn n=0 ' () * k=n (k + 2)(k + 1)ck+2 xk + 3 k=0 ∞ & ∞ & ∞ & ck xk k=0 [(k + 1)kck+1 + 2(k + 2)(k + 1)ck+2 + 3ck ]xk = 0. k=1 Thus 4c2 + 3c0 = 0 (k + 1)kck+1 + 2(k + 2)(k + 1)ck+2 + 3ck = 0 310 CHAPTER 5 REVIEW EXERCISES and 3 c2 = − c0 4 ck+2 = − 3 k ck+1 − ck , 2(k + 2) 2(k + 2)(k + 1) Choosing c0 = 1 and c1 = 0 we find c2 = − k = 1, 2, 3, . . . . 3 4 1 8 1 c4 = 16 c3 = c5 = − 9 320 and so on. For c0 = 0 and c1 = 1 we obtain c2 = 0 c3 = − 1 4 1 16 c5 = 0 c4 = and so on. Thus, the general solution is $ % % $ 3 1 1 1 9 5 1 y = C0 1 − x2 + x3 + x4 − x + · · · + C1 x − x3 + x4 + · · · 4 8 16 320 4 16 and $ % % $ 3 2 1 3 3 2 1 3 9 4 3 ′ y = C0 − x + x + x − x + · · · + C1 1 − x + x + · · · . 2 8 4 64 4 4 Setting y(0) = 0 and y ′ (0) = 1 we find c0 = 0 and c1 = 1. Therefore, the solution of the initial-value problem is 1 1 y = x − x3 + x4 + · · · . 4 16 17. The singular point of (1 − 2 sin x)y ′′ + xy = 0 closest to x = 0 is π/6. Hence a lower bound is π/6. 18. While we can find two solutions of the form y1 = c0 [1 + · · · ] and y2 = c1 [x + · · · ], the initial conditions at x = 1 give solutions for c0 and c1 in terms of infinite series. Letting t = x − 1 the initial-value problem becomes dy d2 y + (t + 1) + y = 0, y(0) = −6, y ′ (0) = 3. dt2 dt "∞ Substituting y = n=0 cn tn into the differential equation, we have ∞ ∞ ∞ ∞ & & & & d2 y dy n−2 n n−1 + (t + 1) + y = n(n − 1)c t + + + nc t nc t cn tn n n n dt2 dt n=2 n=1 n=1 n=0 ' () * ' () * ' () * ' () * k=n−2 = ∞ & k=n (k + 2)(k + 1)ck+2 tk + k=0 = 2c2 + c1 + c0 + ∞ & k=1 ∞ & k=n−1 kck tk + ∞ & k=0 k=n (k + 1)ck+1 tk + ∞ & ck tk k=0 [(k + 2)(k + 1)ck+2 + (k + 1)ck+1 + (k + 1)ck ]tk = 0. k=1 311 CHAPTER 5 REVIEW EXERCISES Thus 2c2 + c1 + c0 = 0 (k + 2)(k + 1)ck+2 + (k + 1)ck+1 + (k + 1)ck = 0 and c1 + c0 2 ck+1 + ck , =− k+2 c2 = − ck+2 k = 1, 2, 3, . . . . Choosing c0 = 1 and c1 = 0 we find 1 c2 = − , 2 c3 = 1 , 6 c4 = 1 , 12 and so on. For c0 = 0 and c1 = 1 we find 1 c3 = − , 6 1 c2 = − , 2 c4 = 1 , 6 and so on. Thus, the general solution is / 0 0 / 1 2 1 3 1 4 1 2 1 3 1 4 y = c0 1 − t + t + t + · · · + c1 t − t − t + t + · · · . 2 6 12 2 6 6 The initial conditions then imply c0 = −6 and c1 = 3. Thus the solution of the initial-value problem is / 0 1 1 1 2 3 4 y = −6 1 − (x − 1) + (x − 1) + (x − 1) + · · · 2 6 12 0 / 1 1 1 2 3 4 + 3 (x − 1) − (x − 1) − (x − 1) + (x − 1) + · · · . 2 6 6 19. Writing the differential equation in the form ′′ y + $ % 1 − cos x ′ y + xy = 0, x and noting that x x3 x5 1 − cos x = − + − ··· x 2 24 720 is analytic at x = 0, we conclude that x = 0 is an ordinary point of the differential equation. 20. Writing the differential equation in the form y ′′ + $ % x y=0 ex − 1 − x and noting that 2 2 x x2 x = − + + − ··· ex − 1 − x x 3 18 270 we see that x = 0 is a singular point of the differential equation. Since $ % x x3 x4 2x2 2 x + + − ··· , = 2x − ex − 1 − x 3 18 270 we conclude that x = 0 is a regular singular point. 312 CHAPTER 5 REVIEW EXERCISES 21. Substituting y = "∞ n n=0 cn x y ′′ + x2 y ′ + 2xy = into the differential equation we have ∞ & n=2 = ' n(n − 1)cn xn−2 + () k=n−2 ∞ & ∞ & n=1 ' * ncn xn+1 + 2 (k + 2)(k + 1)ck+2 xk + ∞ & k=2 = 2c2 + (6c3 + 2c0 )x + ∞ & k=2 cn xn+1 n=0 () * k=n+1 k=0 ∞ & ' () k=n+1 (k − 1)ck−1 xk + 2 * ∞ & ck−1 xk k=1 [(k + 2)(k + 1)ck+2 + (k + 1)ck−1 ]xk = 5 − 2x + 10x3 . Thus, equating coefficients of like powers of x gives 2c2 = 5 6c3 + 2c0 = −2 12c4 + 3c1 = 0 20c5 + 4c2 = 10 (k + 2)(k + 1)ck+2 + (k + 1)ck−1 = 0, k = 4, 5, 6, . . . , and c2 = 5 2 1 1 c3 = − c0 − 3 3 1 c4 = − c1 4 $ % 1 1 1 1 5 c5 = − c2 = − =0 2 5 2 5 2 1 ck−1 . ck+2 = − k+2 Using the recurrence relation, we find 1 1 1 1 c6 = − c3 = (c0 + 1) = 2 c0 + 2 6 3·6 3 · 2! 3 · 2! 1 1 c7 = − c4 = c1 7 4·7 c8 = c11 = c14 = · · · = 0 1 1 1 c9 = − c6 = − 3 c0 − 3 9 3 · 3! 3 · 3! 1 1 c10 = − c7 = − c1 10 4 · 7 · 10 1 1 1 c0 + 4 c12 = − c9 = 4 12 3 · 4! 3 · 4! 1 1 c1 c13 = − c0 = 13 4 · 7 · 10 · 13 313 CHAPTER 5 REVIEW EXERCISES and so on. Thus / 0 1 3 1 1 1 6 9 12 y = c0 1 − x + 2 x − 3 x + 4 x − ··· 3 3 · 2! 3 · 3! 3 · 4! / 0 1 1 7 1 1 + c1 x − x4 + x − x10 + x13 − · · · 4 4·7 4 · 7 · 10 4 · 7 · 10 · 13 0 / 1 1 1 1 5 x6 − 3 x9 + 4 x12 − · · · . + x2 − x3 + 2 2 3 3 · 2! 3 · 3! 3 · 4! 22. (a) From y = − 1 du we obtain u dx 1 d2 u 1 dy =− + 2 dx u dx2 u Then dy/dx = x2 + y 2 becomes − 1 d2 u 1 + 2 2 u dx u $ du dx %2 $ du dx = x2 + %2 1 u2 . $ du dx %2 , d2 u + x2 u = 0. dx2 so (b) The differential equation u′′ + x2 u = 0 has the form (18) in the text with 1 2 2c − 2 = 2 =⇒ c = 2 1 b2 c2 = 1 =⇒ b = 2 1 a2 − p2 c2 = 0 =⇒ p = . 4 1 − 2a = 0 =⇒ a = Then, by (19) in the text, 1/2 u=x % %0 $ / $ 1 2 1 2 x + c2 J−1/4 x . c1 J1/4 2 2 (c) We have / 0 1 du dw dt d 1/2 1 1 1 x1/2 = − 1/2 x w(t) = − 1/2 + x−1/2 w u dx dt dx 2 x w(t) dx x w / 0 0 0 / / 1 dw 1 1 1 dw dw = − 1/2 x3/2 + 1/2 w = − +w =− +w . 2x2 4t dt 2xw dt 2xw dt x w 2x y=− Now 4t so dw d + w = 4t [c1 J1/4 (t) + c2 J−1/4 (t)] + c1 J1/4 (t) + c2 J−1/4 (t) dt dt % %0 $ / $ 1 1 = 4t c1 J−3/4 (t) − J1/4 (t) + c2 − J−1/4 (t) − J3/4 (t) + c1 J1/4 (t) + c2 J−1/4 (t) 4t 4t $ % % $ 1 2 1 2 2 2 = 4c1 tJ−3/4 (t) − 4c2 tJ3/4 (t) = 2c1 x J−3/4 x − 2c2 x J3/4 x , 2 2 y=− 2c1 x2 J−3/4 ( 12 x2 ) − 2c2 x2 J3/4 ( 12 x2 ) −c1 J−3/4 ( 21 x2 ) + c2 J3/4 ( 12 x2 ) = x . 2x[c1 J1/4 ( 12 x2 ) + c2 J−1/4 ( 21 x2 )] c1 J1/4 ( 12 x2 ) + c2 J−1/4 ( 21 x2 ) 314 CHAPTER 5 REVIEW EXERCISES Letting c = c1 /c2 we have y=x J3/4 ( 12 x2 ) − cJ−3/4 ( 12 x2 ) . cJ1/4 ( 21 x2 ) + J−1/4 ( 12 x2 ) 23. (a) Equations (10) and (24) of Section 5.3 in the text imply 9 cos π2 J1/2 (x) − J−1/2 (x) 2 Y1/2 (x) = cos x. = −J−1/2 (x) = − π sin 2 πx (b) From (15) of Section 5.3 in the text I1/2 (x) = i−1/2 J1/2 (ix) I−1/2 (x) = i1/2 J−1/2 (ix) and so I1/2 (x) = 9 ∞ 1 2 & x2n+1 = πx n=0 (2n + 1)! I−1/2 (x) = 9 ∞ 2 & 1 x2n = πx n=0 (2n)! and 9 9 2 sinh x πx 2 cosh x. πx (c) Equation (16) of Section 5.3 in the text and part (b) imply π π I−1/2 (x) − I1/2 (x) K1/2 (x) = = 2 sin π2 2 = 9 79 2 cosh x − πx 9 8 2 sinh x πx / 0 9 ex − e−x π ex + e−x π −x − = e . 2x 2 2 2x 24. (a) Using formula (5) of Section 3.2 in the text, we find that a second solution of (1 − x2 )y ′′ − 2xy ′ = 0 is y2 (x) = 1 · = 3 3 4 e 2x dx/(1−x2 ) dx = 12 dx 1 = ln 2 1−x 2 $ 1+x 1−x 3 % 2 e− ln(1−x ) dx , where partial fractions was used to obtain the last integral. (b) Using formula (5) of Section 3.2 in the text, we find that a second solution of (1 − x2 )y ′′ − 2xy ′ + 2y = 0 is 3 4 e 2x dx/(1−x2 ) 3 2 e− ln(1−x ) dx x2 x2 / $ % 0 $ % 3 1 1+x 1+x dx 1 x dx = x ln − = ln − 1, =x x2 (1 − x2 ) 2 1−x x 2 1−x y2 (x) = x · dx = x where partial fractions was used to obtain the last integral. 315 CHAPTER 5 REVIEW EXERCISES (c) y2 2 y2 2 1 1 1x -1 y2 (x) = 1x -1 -1 -1 -2 -2 1 ln 2 $ 1+x 1−x % y2 = x ln 2 $ 1+x 1−x % −1 25. (a) By the binomial theorem we have + , -.−1/2 1 + t2 − 2xt - (−1/2)(−3/2) 2 (−1/2)(−3/2)(−5/2) 2 1,2 t − 2xt + (t − 2xt)2 + (t − 2xt)3 + · · · =1− 2 2! 3! 1 3 5 = 1 − (t2 − 2xt) + (t2 − 2xt)2 − (t2 − 2xt)3 + · · · 2 8 16 ∞ & 1 1 = 1 + xt + (3x2 − 1)t2 + (5x3 − 3x)t3 + · · · = Pn (x)tn . 2 2 n=0 (b) Letting x = 1 in (1 − 2xt + t2 )−1/2 , we have (1 − 2t + t2 )−1/2 = (1 − t)−1 = 1 = 1 + t + t2 + t 3 + . . . 1−t (|t| < 1) = ∞ & tn . n=0 From part (a) we have ∞ & n=0 Pn (1)tn = (1 − 2t + t2 )−1/2 = ∞ & tn . n=0 Equating the coefficients of corresponding terms in the two series, we see that Pn (1) = 1. Similarly, letting x = −1 we have 1 = 1 − t + t2 − 3t3 + . . . 1+t ∞ ∞ & & Pn (−1)tn , (−1)n tn = = (1 + 2t + t2 )−1/2 = (1 + t)−1 = n=0 n n=0 so that Pn (−1) = (−1) . 316 (|t| < 1) 6 Numerical Solutions of Ordinary Differential Equations EXERCISES 6.1 Euler Methods and Error Analysis 1. h=0.1 xn 1.00 1.10 1.20 1.30 1.40 1.50 3. yn 5.0000 3.9900 3.2546 2.7236 2.3451 2.0801 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 2. h=0.05 xn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 yn 5.0000 4.4475 3.9763 3.5751 3.2342 2.9452 2.7009 2.4952 2.3226 2.1786 2.0592 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 xn 0.00 0.10 0.20 0.30 0.40 0.50 4. h=0.05 yn 0.0000 0.1005 0.2030 0.3098 0.4234 0.5470 h=0.1 yn 0.0000 0.0501 0.1004 0.1512 0.2028 0.2554 0.3095 0.3652 0.4230 0.4832 0.5465 yn 2.0000 1.6600 1.4172 1.2541 1.1564 1.1122 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 317 h=0.05 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 2.0000 1.8150 1.6571 1.5237 1.4124 1.3212 1.2482 1.1916 1.1499 1.1217 1.1056 h=0.05 yn 1.0000 1.1110 1.2515 1.4361 1.6880 2.0488 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 1.0000 1.0526 1.1113 1.1775 1.2526 1.3388 1.4387 1.5556 1.6939 1.8598 2.0619 6.1 5. Euler Methods and Error Analysis h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 7. 9. yn 0.0000 0.0952 0.1822 0.2622 0.3363 0.4053 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 0.0000 0.0488 0.0953 0.1397 0.1823 0.2231 0.2623 0.3001 0.3364 0.3715 0.4054 yn 0.5000 0.5215 0.5362 0.5449 0.5490 0.5503 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 0.5000 0.5116 0.5214 0.5294 0.5359 0.5408 0.5444 0.5469 0.5484 0.5492 0.5495 xn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 yn 1.0000 1.0024 1.0100 1.0228 1.0414 1.0663 1.0984 1.1389 1.1895 1.2526 1.3315 yn 0.0000 0.0050 0.0200 0.0451 0.0805 0.1266 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 0.0000 0.0013 0.0050 0.0113 0.0200 0.0313 0.0451 0.0615 0.0805 0.1022 0.1266 h=0.05 yn 1.0000 1.1079 1.2337 1.3806 1.5529 1.7557 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 318 h=0.05 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 10. h=0.05 yn 1.0000 1.0095 1.0404 1.0967 1.1866 1.3260 h=0.1 xn 0.00 0.10 0.20 0.30 0.40 0.50 8. h=0.05 h=0.1 xn 1.00 1.10 1.20 1.30 1.40 1.50 6. h=0.05 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 1.0000 1.0519 1.1079 1.1684 1.2337 1.3043 1.3807 1.4634 1.5530 1.6503 1.7560 h=0.05 yn 0.5000 0.5250 0.5498 0.5744 0.5986 0.6224 xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 yn 0.5000 0.5125 0.5250 0.5374 0.5498 0.5622 0.5744 0.5866 0.5987 0.6106 0.6224 6.1 Euler Methods and Error Analysis 11. To obtain the analytic solution use the substitution u = x + y − 1. The resulting differential equation in u(x) will be separable. h=0.1 h=0.05 xn yn 0.00 0.10 0.20 0.30 0.40 0.50 12. (a) 2.0000 2.1220 2.3049 2.5858 3.0378 3.8254 Actual Value 2.0000 2.1230 2.3085 2.5958 3.0650 3.9082 xn yn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 2.0000 2.0553 2.1228 2.2056 2.3075 2.4342 2.5931 2.7953 3.0574 3.4057 3.8840 Actual Value 2.0000 2.1230 2.3085 2.5958 3.0650 3.9082 2.5958 2.7997 3.0650 3.4189 3.9082 y (b) 20 15 10 5 1.1 1.2 1.3 1.4 xn 1.00 1.10 1.20 1.30 1.40 Euler 1.0000 1.2000 1.4938 1.9711 2.9060 Imp . Euler 1.0000 1.2469 1.6430 2.4042 4.5085 x 13. (a) Using Euler’s method we obtain y(0.1) ≈ y1 = 1.2. (b) Using y ′′ = 4e2x we see that the local truncation error is y ′′ (c) h2 (0.1)2 = 4e2c = 0.02e2c . 2 2 Since e2x is an increasing function, e2c ≤ e2(0.1) = e0.2 for 0 ≤ c ≤ 0.1. Thus an upper bound for the local truncation error is 0.02e0.2 = 0.0244. (c) Since y(0.1) = e0.2 = 1.2214, the actual error is y(0.1) − y1 = 0.0214, which is less than 0.0244. (d) Using Euler’s method with h = 0.05 we obtain y(0.1) ≈ y2 = 1.21. (e) The error in (d) is 1.2214 − 1.21 = 0.0114. With global truncation error O(h), when the step size is halved we expect the error for h = 0.05 to be one-half the error when h = 0.1. Comparing 0.0114 with 0.214 we see that this is the case. 14. (a) Using the improved Euler’s method we obtain y(0.1) ≈ y1 = 1.22. (b) Using y ′′′ = 8e2x we see that the local truncation error is y ′′′ (c) h3 (0.1)3 = 8e2c = 0.001333e2c . 6 6 Since e2x is an increasing function, e2c ≤ e2(0.1) = e0.2 for 0 ≤ c ≤ 0.1. Thus an upper bound for the local truncation error is 0.001333e0.2 = 0.001628. (c) Since y(0.1) = e0.2 = 1.221403, the actual error is y(0.1) − y1 = 0.001403 which is less than 0.001628. (d) Using the improved Euler’s method with h = 0.05 we obtain y(0.1) ≈ y2 = 1.221025. 319 6.1 Euler Methods and Error Analysis (e) The error in (d) is 1.221403 − 1.221025 = 0.000378. With global truncation error O(h2 ), when the step size is halved we expect the error for h = 0.05 to be one-fourth the error for h = 0.1. Comparing 0.000378 with 0.001403 we see that this is the case. 15. (a) Using Euler’s method we obtain y(0.1) ≈ y1 = 0.8. (b) Using y ′′ = 5e−2x we see that the local truncation error is 5e−2c (0.1)2 = 0.025e−2c . 2 Since e−2x is a decreasing function, e−2c ≤ e0 = 1 for 0 ≤ c ≤ 0.1. Thus an upper bound for the local truncation error is 0.025(1) = 0.025. (c) Since y(0.1) = 0.8234, the actual error is y(0.1) − y1 = 0.0234, which is less than 0.025. (d) Using Euler’s method with h = 0.05 we obtain y(0.1) ≈ y2 = 0.8125. (e) The error in (d) is 0.8234 − 0.8125 = 0.0109. With global truncation error O(h), when the step size is halved we expect the error for h = 0.05 to be one-half the error when h = 0.1. Comparing 0.0109 with 0.0234 we see that this is the case. 16. (a) Using the improved Euler’s method we obtain y(0.1) ≈ y1 = 0.825. (b) Using y ′′′ = −10e−2x we see that the local truncation error is 10e−2c (0.1)3 = 0.001667e−2c . 6 Since e−2x is a decreasing function, e−2c ≤ e0 = 1 for 0 ≤ c ≤ 0.1. Thus an upper bound for the local truncation error is 0.001667(1) = 0.001667. (c) Since y(0.1) = 0.823413, the actual error is y(0.1) − y1 = 0.001587, which is less than 0.001667. (d) Using the improved Euler’s method with h = 0.05 we obtain y(0.1) ≈ y2 = 0.823781. (e) The error in (d) is |0.823413 − 0.8237181| = 0.000305. With global truncation error O(h2 ), when the step size is halved we expect the error for h = 0.05 to be one-fourth the error when h = 0.1. Comparing 0.000305 with 0.001587 we see that this is the case. 17. (a) Using y ′′ = 38e−3(x−1) we see that the local truncation error is y ′′ (c) h2 h2 = 38e−3(c−1) = 19h2 e−3(c−1) . 2 2 (b) Since e−3(x−1) is a decreasing function for 1 ≤ x ≤ 1.5, e−3(c−1) ≤ e−3(1−1) = 1 for 1 ≤ c ≤ 1.5 and y ′′ (c) h2 ≤ 19(0.1)2 (1) = 0.19. 2 (c) Using Euler’s method with h = 0.1 we obtain y(1.5) ≈ 1.8207. With h = 0.05 we obtain y(1.5) ≈ 1.9424. (d) Since y(1.5) = 2.0532, the error for h = 0.1 is E0.1 = 0.2325, while the error for h = 0.05 is E0.05 = 0.1109. With global truncation error O(h) we expect E0.1 /E0.05 ≈ 2. We actually have E0.1 /E0.05 = 2.10. 18. (a) Using y ′′′ = −114e−3(x−1) we see that the local truncation error is ! ! 3 3! ! ′′′ !y (c) h ! = 114e−3(x−1) h = 19h3 e−3(c−1) . ! ! 6 6 (b) Since e−3(x−1) is a decreasing function for 1 ≤ x ≤ 1.5, e−3(c−1) ≤ e−3(1−1) = 1 for 1 ≤ c ≤ 1.5 and ! ! 3! ! ′′′ !y (c) h ! ≤ 19(0.1)3 (1) = 0.019. ! 6! 320 6.2 Runge-Kutta Methods (c) Using the improved Euler’s method with h = 0.1 we obtain y(1.5) ≈ 2.080108. With h = 0.05 we obtain y(1.5) ≈ 2.059166. (d) Since y(1.5) = 2.053216, the error for h = 0.1 is E0.1 = 0.026892, while the error for h = 0.05 is E0.05 = 0.005950. With global truncation error O(h2 ) we expect E0.1 /E0.05 ≈ 4. We actually have E0.1 /E0.05 = 4.52. 19. (a) Using y ′′ = −1/(x + 1)2 we see that the local truncation error is ! ! 2! ! ′′ 1 h2 !y (c) h ! = . ! 2 ! (c + 1)2 2 (b) Since 1/(x + 1)2 is a decreasing function for 0 ≤ x ≤ 0.5, 1/(c + 1)2 ≤ 1/(0 + 1)2 = 1 for 0 ≤ c ≤ 0.5 and ! ! 2 2! ! ′′ !y (c) h ! ≤ (1) (0.1) = 0.005. ! 2! 2 (c) Using Euler’s method with h = 0.1 we obtain y(0.5) ≈ 0.4198. With h = 0.05 we obtain y(0.5) ≈ 0.4124. (d) Since y(0.5) = 0.4055, the error for h = 0.1 is E0.1 = 0.0143, while the error for h = 0.05 is E0.05 = 0.0069. With global truncation error O(h) we expect E0.1 /E0.05 ≈ 2. We actually have E0.1 /E0.05 = 2.06. 20. (a) Using y ′′′ = 2/(x + 1)3 we see that the local truncation error is y ′′′ (c) 1 h3 h3 = . 6 (c + 1)3 3 (b) Since 1/(x + 1)3 is a decreasing function for 0 ≤ x ≤ 0.5, 1/(c + 1)3 ≤ 1/(0 + 1)3 = 1 for 0 ≤ c ≤ 0.5 and y ′′′ (c) (0.1)3 h3 ≤ (1) = 0.000333. 6 3 (c) Using the improved Euler’s method with h = 0.1 we obtain y(0.5) ≈ 0.405281. With h = 0.05 we obtain y(0.5) ≈ 0.405419. (d) Since y(0.5) = 0.405465, the error for h = 0.1 is E0.1 = 0.000184, while the error for h = 0.05 is E0.05 = 0.000046. With global truncation error O(h2 ) we expect E0.1 /E0.05 ≈ 4. We actually have E0.1 /E0.05 = 3.98. ∗ depends on yn and is used to determine yn+1 , all of the yn∗ cannot be computed at one time 21. Because yn+1 independently of the corresponding yn values. For example, the computation of y4∗ involves the value of y3 . EXERCISES 6.2 Runge-Kutta Methods 1. xn 0.00 0.10 0.20 0.30 0.40 0.50 yn 2.0000 2.1230 2.3085 2.5958 3.0649 3.9078 Actual Value 2.0000 2.1230 2.3085 2.5958 3.0650 3.9082 321 6.2 Runge-Kutta Methods 2. In this problem we use h = 0.1. Substituting w2 = 3 4 into the equations xn in (4) in the text, we obtain w1 = 1 − w2 = 1 , 4 α= 2 1 = , 2w2 3 and β = 0.00 0.10 0.20 0.30 0.40 0.50 2 1 = . 2w2 3 The resulting second-order Runge-Kutta method is yn+1 = yn + h " 1 3 k1 + k2 4 4 # = yn + Second !Order Improved Runge !Kutta Euler 2.0000 2.0000 2.1213 2.1220 2.3030 2.3049 2.5814 2.5858 3.0277 3.0378 3.8002 3.8254 h (k1 + 3k2 ) 4 where k1 = f (xn , yn ) " # 2 2 k2 = f xn + h, yn + hk1 . 3 3 The table compares the values obtained using this second-order Runge-Kutta method with the values obtained using the improved Euler’s method. 3. xn 1.00 1.10 1.20 1.30 1.40 1.50 5. xn 0.00 0.10 0.20 0.30 0.40 0.50 7. xn 0.00 0.10 0.20 0.30 0.40 0.50 9. xn 0.00 0.10 0.20 0.30 0.40 0.50 yn 4. 5.0000 3.9724 3.2284 2.6945 2.3163 2.0533 yn 0.00 0.10 0.20 0.30 0.40 0.50 6. 0.0000 0.1003 0.2027 0.3093 0.4228 0.5463 yn xn 0.00 0.10 0.20 0.30 0.40 0.50 8. 0.0000 0.0953 0.1823 0.2624 0.3365 0.4055 yn xn xn 0.00 0.10 0.20 0.30 0.40 0.50 10. 0.5000 0.5213 0.5358 0.5443 0.5482 0.5493 xn 0.00 0.10 0.20 0.30 0.40 0.50 322 yn 2.0000 1.6562 1.4110 1.2465 1.1480 1.1037 yn 1.0000 1.1115 1.2530 1.4397 1.6961 2.0670 yn 0.0000 0.0050 0.0200 0.0451 0.0805 0.1266 yn 1.0000 1.1079 1.2337 1.3807 1.5531 1.7561 6.2 xn 11. yn 1.00 1.10 1.20 1.30 1.40 1.50 12. 1.0000 1.0101 1.0417 1.0989 1.1905 1.3333 13. (a) Write the equation in the form dv = 32 − 0.125v 2 = f (t, v). dt (b) v 40 xn Runge-Kutta Methods yn 0.00 0.10 0.20 0.30 0.40 0.50 0.5000 0.5250 0.5498 0.5744 0.5987 0.6225 tn 0.0 1.0 2.0 3.0 4.0 5.0 vn 0.0000 25.2570 32.9390 34.9770 35.5500 35.7130 30 20 10 1 2 3 4 5 6 t (c) Separating variables and using partial fractions we have " # 1 1 1 √ √ √ √ dv = dt +√ 2 32 32 − 0.125 v 32 + 0.125 v and √ 1 √ $ % √ √ √ √ ln | 32 + 0.125 v| − ln | 32 − 0.125 v| = t + c. 2 32 0.125 Since v(0) = 0 we find c = 0. Solving for v we obtain √ √ 16 5 (e 3.2 t − 1) √ v(t) = e 3.2 t + 1 and v(5) ≈ 35.7678. Alternatively, the solution can be expressed as & & mg kg v(t) = tanh t. k m 14. (a) t !days " A !observed " A !approximated " 1 2.78 1.93 2 13.53 12.50 3 36.30 36.46 4 47.50 47.23 (b) From the graph we estimate A(1) ≈ 1.68, A(2) ≈ 13.2, A(3) ≈ 36.8, A(4) ≈ 46.9, and A(5) ≈ 48.9. 5 49.40 49.00 A(t) 50 40 30 20 10 0 323 1 2 3 4 5 t 6.2 Runge-Kutta Methods (c) Let α = 2.128 and β = 0.0432. Separating variables we obtain dA = dt A(α − βA) " # 1 1 β + dA = dt α A α − βA 1 [ln A − ln(α − βA)] = t + c α A ln = α(t + c) α − βA A = eα(t+c) α − βA A = αeα(t+c) − βAeα(t+c) ' ( 1 + βeα(t+c) A = αeα(t+c) . Thus A(t) = αeα(t+c) α α = = . β + e−αc e−αt 1 + βeα(t+c) β + e−α(t+c) From A(0) = 0.24 we obtain 0.24 = α β + e−αc so that e−αc = α/0.24 − β ≈ 8.8235 and A(t) ≈ t !days " A !observed " A !actual " 15. (a) 1 2.78 1.93 xn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 h"0.05 1.0000 1.1112 1.2511 1.4348 1.6934 2.1047 2.9560 7.8981 1.40 1.0608 ! 1015 2 13.53 12.50 2.128 . 0.0432 + 8.8235e−2.128t 3 36.30 36.46 4 47.50 47.23 5 49.40 49.00 (b) h"0.1 1.0000 y 20 1.2511 15 1.6934 10 5 2.9425 1.1 903.0282 16. (a) Using the RK4 method we obtain y(0.1) ≈ y1 = 1.2214. (b) Using y (5) (x) = 32e2x we see that the local truncation error is y (5) (c) (0.1)5 h5 = 32e2c = 0.000002667e2c . 120 120 324 1.2 1.3 1.4 x 6.2 Runge-Kutta Methods Since e2x is an increasing function, e2c ≤ e2(0.1) = e0.2 for 0 ≤ c ≤ 0.1. Thus an upper bound for the local truncation error is 0.000002667e0.2 = 0.000003257. (c) Since y(0.1) = e0.2 = 1.221402758, the actual error is y(0.1) − y1 = 0.000002758 which is less than 0.000003257. (d) Using the RK4 formula with h = 0.05 we obtain y(0.1) ≈ y2 = 1.221402571. (e) The error in (d) is 1.221402758 − 1.221402571 = 0.000000187. With global truncation error O(h4 ), when the step size is halved we expect the error for h = 0.05 to be one-sixteenth the error for h = 0.1. Comparing 0.000000187 with 0.000002758 we see that this is the case. 17. (a) Using the RK4 method we obtain y(0.1) ≈ y1 = 0.823416667. (b) Using y (5) (x) = −40e−2x we see that the local truncation error is 40e−2c (0.1)5 = 0.000003333. 120 Since e−2x is a decreasing function, e−2c ≤ e0 = 1 for 0 ≤ c ≤ 0.1. Thus an upper bound for the local truncation error is 0.000003333(1) = 0.000003333. (c) Since y(0.1) = 0.823413441, the actual error is |y(0.1) − y1 | = 0.000003225, which is less than 0.000003333. (d) Using the RK4 method with h = 0.05 we obtain y(0.1) ≈ y2 = 0.823413627. (e) The error in (d) is |0.823413441 − 0.823413627| = 0.000000185. With global truncation error O(h4 ), when the step size is halved we expect the error for h = 0.05 to be one-sixteenth the error when h = 0.1. Comparing 0.000000185 with 0.000003225 we see that this is the case. 18. (a) Using y (5) = −1026e−3(x−1) we see that the local truncation error is ! ! 5 ! ! (5) !y (c) h ! = 8.55h5 e−3(c−1) . ! 120 ! (b) Since e−3(x−1) is a decreasing function for 1 ≤ x ≤ 1.5, e−3(c−1) ≤ e−3(1−1) = 1 for 1 ≤ c ≤ 1.5 and y (5) (c) h5 ≤ 8.55(0.1)5 (1) = 0.0000855. 120 (c) Using the RK4 method with h = 0.1 we obtain y(1.5) ≈ 2.053338827. With h = 0.05 we obtain y(1.5) ≈ 2.053222989. 19. (a) Using y (5) = 24/(x + 1)5 we see that the local truncation error is y (5) (c) h5 h5 1 = . 5 120 (c + 1) 5 (b) Since 1/(x + 1)5 is a decreasing function for 0 ≤ x ≤ 0.5, 1/(c + 1)5 ≤ 1/(0 + 1)5 = 1 for 0 ≤ c ≤ 0.5 and y (5) (c) (0.1)5 h5 ≤ (1) = 0.000002. 5 5 (c) Using the RK4 method with h = 0.1 we obtain y(0.5) ≈ 0.405465168. With h = 0.05 we obtain y(0.5) ≈ 0.405465111. 20. Each step of Euler’s method requires only 1 function evaluation, while each step of the improved Euler’s method ∗ requires 2 function evaluations – once at (xn , yn ) and again at (xn+1 , yn+1 ). The second-order Runge-Kutta methods require 2 function evaluations per step, while the RK4 method requires 4 function evaluations per step. To compare the methods we approximate the solution of y ′ = (x + y − 1)2 , y(0) = 2, at x = 0.2 using h = 0.1 325 6.2 Runge-Kutta Methods for the Runge-Kutta method, h = 0.05 for the improved Euler’s method, and h = 0.025 for Euler’s method. For each method a total of 8 function evaluations is required. By comparing with the exact solution we see that the RK4 method appears to still give the most accurate result. xn 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Imp . Euler Euler h"0.025 h"0.05 2.0000 2.0000 2.0250 2.0526 2.0553 2.0830 2.1165 2.1228 2.1535 2.1943 2.2056 2.2395 2.2895 2.3075 RK4 h"0.1 2.0000 2.1230 2.3085 Actual 2.0000 2.0263 2.0554 2.0875 2.1230 2.1624 2.2061 2.2546 2.3085 21. (a) For y ′ + y = 10 sin 3x an integrating factor is ex so that y 5 d x [e y] = 10ex sin 3x =⇒ ex y = ex sin 3x − 3ex cos 3x + c dx =⇒ y = sin 3x − 3 cos 3x + ce−x . When x = 0, y = 0, so 0 = −3 + c and c = 3. The solution is 2 x y = sin 3x − 3 cos 3x + 3e−x . Using Newton’s method we find that x = 1.53235 is the only positive root in [0, 2]. −5 (b) Using the RK4 method with h = 0.1 we obtain the table of values shown. These values are used to obtain an interpolating function in Mathematica. The graph of the interpolating function is shown. Using Mathematica’s root finding capability we see that the only positive root in [0, 2] is x = 1.53236. xn 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 yn 0.0000 0.1440 0.5448 1.1409 1.8559 2.6049 3.3019 3.8675 4.2356 4.3593 4.2147 xn 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 yn y 4.2147 3.8033 3.1513 2.3076 1.3390 0.3243 "0.6530 "1.5117 "2.1809 "2.6061 "2.7539 5 2 −5 326 x 6.3 Multistep Methods EXERCISES 6.3 Multistep Methods In the tables in this section “ABM” stands for Adams-Bashforth-Moulton. 1. Writing the differential equation in the form y ′ − y = x − 1 we see that an integrating factor is e− so that d −x [e y] = (x − 1)e−x dx and y = ex (−xe−x + c) = −x + cex . ) dx = e−x , From y(0) = 1 we find c = 1, so the solution of the initial-value problem is y = −x + ex . Actual values of the analytic solution above are compared with the approximated values in the table. xn 0.0 0.2 0.4 0.6 0.8 yn 1.00000000 1.02140000 1.09181796 1.22210646 1.42552788 Actual 1.00000000 1.02140276 1.09182470 1.22211880 1.42554093 init. cond. RK4 RK4 RK4 ABM 2. The following program is written in Mathematica. It uses the Adams-Bashforth-Moulton method to approximate the solution of the initial-value problem y ′ = x + y − 1, y(0) = 1, on the interval [0, 1]. Clear[f, x, y, h, a, b, y0]; f[x , y ]:= x + y - 1; h = 0.2; (* define the differential equation *) (* set the step size *) a = 0; y0 = 1; b = 1; f[x, y] (* set the initial condition and the interval *) (* display the DE *) Clear[k1, k2, k3, k4, x, y, u, v] x = u[0] = a; y = v[0] = y0; n = 0; While[x < a + 3h, (* use RK4 to compute the first 3 values after y(0) *) n = n + 1; k1 = f[x, y]; k2 = f[x + h/2, y + h k1/2]; k3 = f[x + h/2, y + h k2/2]; k4 = f[x + h, y + h k3]; x = x + h; y = y + (h/6)(k1 + 2k2 + 2k3 + k4); u[n] = x; v[n] = y]; 327 6.3 Multistep Methods While[x ≤ b, (* use Adams-Bashforth-Moulton *) p3 = f[u[n - 3], v[n - 3]]; p2 = f[u[n - 2], v[n - 2]]; p1 = f[u[n - 1], v[n - 1]]; p0 = f[u[n], v[n]]; pred = y + (h/24)(55p0 - 59p1 + 37p2 - 9p3); (* predictor *) x = x + h; p4 = f[x, pred]; y = y + (h/24)(9p4 + 19p0 - 5p1 + p2); (* corrector *) n = n + 1; u[n] = x; v[n] = y] (*display the table *) TableForm[Prepend[Table[{u[n], v[n]}, {n, 0, (b-a)/h}], {"x(n)", "y(n)"}]]; 3. The first predictor is y4∗ = 0.73318477. xn 0.0 0.2 0.4 0.6 0.8 yn 1.00000000 0.73280000 0.64608032 0.65851653 0.72319464 init. cond. RK4 RK4 RK4 ABM 4. The first predictor is y4∗ = 1.21092217. xn 0.0 0.2 0.4 0.6 0.8 yn 2.00000000 1.41120000 1.14830848 1.10390600 1.20486982 init. cond. RK4 RK4 RK4 ABM 5. The first predictor for h = 0.2 is y4∗ = 1.02343488. xn 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 h"0.2 0.00000000 init. cond. 0.20270741 RK4 0.42278899 RK4 0.68413340 RK4 1.02969040 ABM 1.55685960 ABM h"0.1 0.00000000 0.10033459 0.20270988 0.30933604 0.42279808 0.54631491 0.68416105 0.84233188 1.02971420 1.26028800 1.55762558 init. cond. RK4 RK4 RK4 ABM ABM ABM ABM ABM ABM ABM 328 6.3 6. The first predictor for h = 0.2 is y4∗ = 3.34828434. xn 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 h"0.2 1.00000000 init. cond. 1.44139950 RK4 1.97190167 RK4 2.60280694 RK4 3.34860927 ABM 4.22797875 ABM h"0.1 1.00000000 1.21017082 1.44140511 1.69487942 1.97191536 2.27400341 2.60283209 2.96031780 3.34863769 3.77026548 4.22801028 init. cond. RK4 RK4 RK4 ABM ABM ABM ABM ABM ABM ABM 7. The first predictor for h = 0.2 is y4∗ = 0.13618654. xn 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 h"0.2 0.00000000 init. cond. 0.00262739 RK4 0.02005764 RK4 0.06296284 RK4 0.13598600 ABM 0.23854783 ABM h"0.1 0.00000000 0.00033209 0.00262486 0.00868768 0.02004821 0.03787884 0.06294717 0.09563116 0.13596515 0.18370712 0.23841344 init. cond. RK4 RK4 RK4 ABM ABM ABM ABM ABM ABM ABM 8. The first predictor for h = 0.2 is y4∗ = 2.61796154. xn 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 h"0.2 1.00000000 init. cond. 1.23369623 RK4 1.55308554 RK4 1.99610329 RK4 2.62136177 ABM 3.52079042 ABM h"0.1 1.00000000 1.10793839 1.23369772 1.38068454 1.55309381 1.75610064 1.99612995 2.28119129 2.62131818 3.02914333 3.52065536 init. cond. RK4 RK4 RK4 ABM ABM ABM ABM ABM ABM ABM 329 Multistep Methods 6.3 Higher-Order Multistep Methods 6.4 Equations and Systems EXERCISES 6.4 Higher-Order Equations and Systems 1. The substitution y ′ = u leads to the iteration formulas yn+1 = yn + hun , un+1 = un + h(4un − 4yn ). The initial conditions are y0 = −2 and u0 = 1. Then y1 = y0 + 0.1u0 = −2 + 0.1(1) = −1.9 u1 = u0 + 0.1(4u0 − 4y0 ) = 1 + 0.1(4 + 8) = 2.2 y2 = y1 + 0.1u1 = −1.9 + 0.1(2.2) = −1.68. The general solution of the differential equation is y = c1 e2x + c2 xe2x . From the initial conditions we find c1 = −2 and c2 = 5. Thus y = −2e2x + 5xe2x and y(0.2) ≈ 1.4918. 2. The substitution y ′ = u leads to the iteration formulas yn+1 = yn + hun , un+1 = un + h " # 2 2 u n − 2 yn . x x The initial conditions are y0 = 4 and u0 = 9. Then y1 = y0 + 0.1u0 = 4 + 0.1(9) = 4.9 " # 2 2 u1 = u0 + 0.1 u0 − y0 = 9 + 0.1[2(9) − 2(4)] = 10 1 1 y2 = y1 + 0.1u1 = 4.9 + 0.1(10) = 5.9. The general solution of the Cauchy-Euler differential equation is y = c1 x + c2 x2 . From the initial conditions we find c1 = −1 and c2 = 5. Thus y = −x + 5x2 and y(1.2) = 6. 3. The substitution y ′ = u leads to the system ′ y = u, xn ′ u = 4u − 4y. Using formula (4) in the text with x corresponding to t, y corresponding to x, and u corresponding to y, we obtain the table shown. 0.0 0.1 0.2 4. The substitution y ′ = u leads to the system y ′ = u, 2 2 u′ = u − 2 y. x x Using formula (4) in the text with x corresponding to t, y corresponding to x, and u corresponding to y, we obtain the table shown. 330 h"0.2 yn h"0.2 un "2.0000 1.0000 "1.4928 4.4731 h"0.1 yn "2.0000 "1.8321 "1.4919 xn h"0.2 yn h"0.2 un h"0.1 yn 1.0 1.1 1.2 4.0000 9.0000 6.0001 11.0002 4.0000 4.9500 6.0000 h"0.1 un 1.0000 2.4427 4.4753 h"0.1 un 9.0000 10.0000 11.0000 6.4 5. The substitution y ′ = u leads to the system y ′ = u, u′ = 2u − 2y + et cos t. Using formula (4) in the text with y corresponding to x and u corresponding to y, we obtain the table shown. Higher-Order Equations and Systems xn h"0.2 yn h"0.2 un h"0.1 yn h"0.1 un 0.0 0.1 0.2 1.0000 2.0000 1.4640 2.6594 1.0000 1.2155 1.4640 2.0000 2.3150 2.6594 6. Using h = 0.1, the RK4 method for a system, and a numerical solver, we obtain tn 0.0 0.1 0.2 0.3 0.4 0.5 7. h"0.2 i 1n 0.0000 2.5000 2.8125 2.0703 0.6104 "1.5619 i1 7 h"0.2 i 3n 0.0000 3.7500 5.7813 7.4023 9.1919 11.4877 i2 7 6 6 5 5 4 4 3 3 2 2 1 1 1 tn h"0.2 xn h"0.2 yn h"0.1 xn h"0.1 yn 0.0 0.1 0.2 6.0000 2.0000 8.3055 3.4199 6.0000 7.0731 8.3055 2.0000 2.6524 3.4199 2 3 4 5t 1 2 3 4 5t x,y 20 xHtL 15 10 yHtL 5 0.5 8. 1 1.5 2t 1.5 2t x,y tn h"0.2 xn h"0.2 yn h"0.1 xn h"0.1 yn 50 0.0 0.1 0.2 1.0000 1.0000 1.0000 1.4006 2.0845 1.0000 1.8963 3.3502 40 2.0785 3.3382 yHtL 30 20 10 xHtL 0.5 331 1 6.4 Higher-Order Equations and Systems x,y 9. tn 0.0 0.1 0.2 h"0.2 xn h"0.2 yn "3.0000 5.0000 "3.9123 4.2857 h"0.1 xn h"0.1 yn "3.0000 "3.4790 "3.9123 30 5.0000 4.6707 4.2857 25 20 15 xHtL 10 yHtL 5 5 10 15 20 25 30 t -5 10. tn h"0.2 xn h"0.2 yn h"0.1 xn h"0.1 yn 0.0 0.1 0.2 0.5000 0.2000 2.1589 2.3279 0.5000 1.0207 2.1904 0.2000 1.0115 2.3592 x,y 2 xHtL 1.5 1 0.5 yHtL 0.1 11. Solving for x′ and y ′ we obtain the system 1 y ′ = 2x + y − 2t. -5 h"0.2 yn tn 0.0 0.1 0.2 1.0000 "2.0000 0.4179 "2.1824 h"0.1 xn 1.0000 0.6594 0.4173 0.3t 3 4t x,y x′ = −2x + y + 5t h"0.2 xn 0.2 h"0.1 yn xHtL -10 "2.0000 "2.0476 "2.1821 -15 yHtL -20 332 2 6.5 Second-Order Boundary-Value Problems 12. Solving for x′ and y ′ we obtain the system 1 x = y − 3t2 + 2t − 5 2 1 ′ y = − y + 3t2 + 2t + 5. 2 ′ x,y 60 40 yHtL 20 tn h"0.2 xn h"0.2 yn 0.0 0.1 0.2 3.0000 "1.0000 1.9867 0.0933 h"0.1 xn h"0.1 yn 3.0000 2.4727 1.9867 2 -20 "1.0000 "0.4527 0.0933 4 6 8t xHtL -40 -60 EXERCISES 6.5 Second-Order Boundary-Value Problems 1. We identify P (x) = 0, Q(x) = 9, f (x) = 0, and h = (2 − 0)/4 = 0.5. Then the finite difference equation is yi+1 + 0.25yi + yi−1 = 0. The solution of the corresponding linear system gives x y 0.0 0.5 1.0 1.5 4.0000 -5.6774 -2.5807 6.3226 2.0 1.0000 2. We identify P (x) = 0, Q(x) = −1, f (x) = x2 , and h = (1 − 0)/4 = 0.25. Then the finite difference equation is yi+1 − 2.0625yi + yi−1 = 0.0625x2i . The solution of the corresponding linear system gives x y 0.00 0.25 0.50 0.75 1.00 0.0000 -0.0172 -0.0316 -0.0324 0.0000 3. We identify P (x) = 2, Q(x) = 1, f (x) = 5x, and h = (1 − 0)/5 = 0.2. Then the finite difference equation is 1.2yi+1 − 1.96yi + 0.8yi−1 = 0.04(5xi ). The solution of the corresponding linear system gives x y 0.0 0.2 0.4 0.6 0.8 1.0 0.0000 -0.2259 -0.3356 -0.3308 -0.2167 0.0000 333 6.5 Second-Order Boundary-Value Problems 4. We identify P (x) = −10, Q(x) = 25, f (x) = 1, and h = (1 − 0)/5 = 0.2. Then the finite difference equation is −yi + 2yi−1 = 0.04. The solution of the corresponding linear system gives x y 0.0 1.0000 0.2 1.9600 0.4 3.8800 0.6 0.8 1.0 7.7200 15.4000 0.0000 5. We identify P (x) = −4, Q(x) = 4, f (x) = (1 + x)e2x , and h = (1 − 0)/6 = 0.1667. Then the finite difference equation is 0.6667yi+1 − 1.8889yi + 1.3333yi−1 = 0.2778(1 + xi )e2xi . The solution of the corresponding linear system gives x y 0.0000 3.0000 0.1667 3.3751 0.3333 3.6306 0.5000 3.6448 0.6667 3.2355 0.8333 2.1411 1.0000 0.0000 √ 6. We identify P (x) = 5, Q(x) = 0, f (x) = 4 x , and h = (2 − 1)/6 = 0.1667. Then the finite difference equation is √ 1.4167yi+1 − 2yi + 0.5833yi−1 = 0.2778(4 xi ). The solution of the corresponding linear system gives x y 1.0000 1.1667 1.3333 1.5000 1.6667 1.8333 2.0000 1.0000 -0.5918 -1.1626 -1.3070 -1.2704 -1.1541 -1.0000 7. We identify P (x) = 3/x, Q(x) = 3/x2 , f (x) = 0, and h = (2 − 1)/8 = 0.125. Then the finite difference equation is " 0.1875 1+ xi # " 0.0469 yi+1 + −2 + x2i # " 0.1875 yi + 1 − xi # yi−1 = 0. The solution of the corresponding linear system gives x y 1.000 5.0000 1.125 3.8842 1.250 2.9640 1.375 2.2064 1.500 1.5826 1.625 1.0681 1.750 0.6430 1.875 0.2913 2.000 0.0000 8. We identify P (x) = −1/x, Q(x) = x−2 , f (x) = ln x/x2 , and h = (2 − 1)/8 = 0.125. Then the finite difference equation is # " # # " " 0.0156 0.0625 0.0625 yi+1 + −2 + yi−1 = 0.0156 ln xi . yi + 1 + 1− xi x2i xi The solution of the corresponding linear system gives x y 1.000 1.125 1.250 1.375 1.500 1.625 1.750 1.875 2.000 0.0000 -0.1988 -0.4168 -0.6510 -0.8992 -1.1594 -1.4304 -1.7109 -2.0000 9. We identify P (x) = 1 − x, Q(x) = x, f (x) = x, and h = (1 − 0)/10 = 0.1. Then the finite difference equation is [1 + 0.05(1 − xi )]yi+1 + [−2 + 0.01xi ]yi + [1 − 0.05(1 − xi )]yi−1 = 0.01xi . The solution of the corresponding linear system gives x y 0.0 0.0000 0.1 0.2660 0.2 0.5097 0.3 0.7357 0.4 0.9471 0.5 0.6 1.1465 1.3353 0.7 0.8 1.5149 1.6855 0.9 1.8474 1.0 2.0000 10. We identify P (x) = x, Q(x) = 1, f (x) = x, and h = (1 − 0)/10 = 0.1. Then the finite difference equation is (1 + 0.05xi )yi+1 − 1.99yi + (1 − 0.05xi )yi−1 = 0.01xi . 334 6.5 Second-Order Boundary-Value Problems The solution of the corresponding linear system gives x y 0.0 1.0000 0.1 0.8929 0.2 0.7789 0.3 0.6615 0.4 0.5440 0.5 0.6 0.4296 0.3216 0.7 0.8 0.2225 0.1347 0.9 0.0601 1.0 0.0000 11. We identify P (x) = 0, Q(x) = −4, f (x) = 0, and h = (1 − 0)/8 = 0.125. Then the finite difference equation is yi+1 − 2.0625yi + yi−1 = 0. The solution of the corresponding linear system gives x y 0.000 0.0000 0.125 0.3492 0.250 0.7202 0.375 1.1363 0.500 1.6233 0.625 2.2118 0.750 2.9386 0.875 3.8490 1.000 5.0000 12. We identify P (r) = 2/r, Q(r) = 0, f (r) = 0, and h = (4 − 1)/6 = 0.5. Then the finite difference equation is " # " # 0.5 0.5 1+ ui+1 − 2ui + 1 − ui−1 = 0. ri ri The solution of the corresponding linear system gives r 1.0 1.5 2.0 2.5 3.0 3.5 4.0 u 50.0000 72.2222 83.3333 90.0000 94.4444 97.6190 100.0000 13. (a) The difference equation # " # " h h 2 1 + Pi yi+1 + (−2 + h Qi )yi + 1 − Pi yi−1 = h2 fi 2 2 is the same as equation (8) in the text. The equations are the same because the derivation was based only on the differential equation, not the boundary conditions. If we allow i to range from 0 to n − 1 we obtain n equations in the n + 1 unknowns y−1 , y0 , y1 , . . . , yn−1 . Since yn is one of the given boundary conditions, it is not an unknown. (b) Identifying y0 = y(0), y−1 = y(0 − h), and y1 = y(0 + h) we have from equation (5) in the text 1 [y1 − y−1 ] = y ′ (0) = 1 or y1 − y−1 = 2h. 2h The difference equation corresponding to i = 0, " " # # h h 1 + P0 y1 + (−2 + h2 Q0 )y0 + 1 − P0 y−1 = h2 f0 2 2 becomes, with y−1 = y1 − 2h, " # " # h h 2 1 + P0 y1 + (−2 + h Q0 )y0 + 1 − P0 (y1 − 2h) = h2 f0 2 2 or 2y1 + (−2 + h2 Q0 )y0 = h2 f0 + 2h − P0 . Alternatively, we may simply add the equation y1 − y−1 = 2h to the list of n difference equations obtaining n + 1 equations in the n + 1 unknowns y−1 , y0 , y1 , . . . , yn−1 . (c) Using n = 5 we obtain x 0.0 0.2 0.4 0.6 0.8 1.0 y -2.2755 -2.0755 -1.8589 -1.6126 -1.3275 -1.0000 335 6.5 Second-Order Boundary-Value Problems 14. Using h = 0.1 and, after shooting a few times, y ′ (0) = 0.43535 we obtain the following table with the RK4 method. x y 0.0 1.00000 0.1 1.04561 0.2 1.09492 0.3 1.14714 0.4 1.20131 0.5 1.25633 0.6 1.31096 0.7 1.36392 0.8 1.41388 0.9 1.45962 CHAPTER 6 REVIEW EXERCISES 1. xn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 2. xn 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Euler h"0.1 2.0000 2.1386 2.3097 2.5136 2.7504 3.0201 Euler h"0.1 0.0000 0.1000 0.2010 0.3049 0.4135 0.5279 Euler Imp . Euler Imp . Euler h"0.05 h"0.1 h"0.05 2.0000 2.0000 2.0000 2.0693 2.0735 2.1469 2.1549 2.1554 2.2328 2.2459 2.3272 2.3439 2.3450 2.4299 2.4527 2.5409 2.5672 2.5689 2.6604 2.6937 2.7883 2.8246 2.8269 2.9245 2.9686 3.0690 3.1157 3.1187 RK4 h"0.1 2.0000 Euler Imp . Euler Imp . Euler h"0.05 h"0.1 h"0.05 0.0000 0.0000 0.0000 0.0500 0.0501 0.1001 0.1005 0.1004 0.1506 0.1512 0.2017 0.2030 0.2027 0.2537 0.2552 0.3067 0.3092 0.3088 0.3610 0.3638 0.4167 0.4207 0.4202 0.4739 0.4782 0.5327 0.5382 0.5378 RK4 h"0.1 0.0000 2.1556 2.3454 2.5695 2.8278 3.1197 0.1003 0.2026 0.3087 0.4201 0.5376 336 RK4 h"0.05 2.0000 2.0736 2.1556 2.2462 2.3454 2.4532 2.5695 2.6944 2.8278 2.9696 3.1197 RK4 h"0.05 0.0000 0.0500 0.1003 0.1511 0.2026 0.2551 0.3087 0.3637 0.4201 0.4781 0.5376 1.0 1.50003 CHAPTER 6 REVIEW EXERCISES 3. xn 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 4. xn 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 Imp . Euler Imp . Euler Euler h"0.05 h"0.1 h"0.05 0.5000 0.5000 0.5000 0.5500 0.5512 0.6024 0.6048 0.6049 0.6573 0.6609 0.7144 0.7191 0.7193 0.7739 0.7800 0.8356 0.8427 0.8430 0.8996 0.9082 0.9657 0.9752 0.9755 1.0340 1.0451 1.1044 1.1163 1.1168 Euler h"0.1 0.5000 0.6000 0.7095 0.8283 0.9559 1.0921 Euler h"0.1 1.0000 1.2000 1.4760 1.8710 2.4643 3.4165 Euler Imp . Euler Imp . Euler h"0.05 h"0.1 h"0.05 1.0000 1.0000 1.0000 1.1000 1.1091 1.2183 1.2380 1.2405 1.3595 1.4010 1.5300 1.5910 1.6001 1.7389 1.8523 1.9988 2.1524 2.1799 2.3284 2.6197 2.7567 3.1458 3.2360 3.3296 4.1528 4.1253 5.2510 5.6404 RK4 h"0.1 0.5000 0.6049 0.7194 0.8431 0.9757 1.1169 RK4 h"0.1 1.0000 1.2415 1.6036 2.1909 3.2745 5.8338 RK4 h"0.05 0.5000 0.5512 0.6049 0.6610 0.7194 0.7801 0.8431 0.9083 0.9757 1.0452 1.1169 RK4 h"0.05 1.0000 1.1095 1.2415 1.4029 1.6036 1.8586 2.1911 2.6401 3.2755 4.2363 5.8446 5. Using yn+1 = yn + hun , y0 = 3 un+1 = un + h(2xn + 1)yn , u0 = 1 we obtain (when h = 0.2) y1 = y(0.2) = y0 + hu0 = 3 + (0.2)1 = 3.2. When h = 0.1 we have y1 = y0 + 0.1u0 = 3 + (0.1)1 = 3.1 u1 = u0 + 0.1(2x0 + 1)y0 = 1 + 0.1(1)3 = 1.3 y2 = y1 + 0.1u1 = 3.1 + 0.1(1.3) = 3.23. 6. The first predictor is y3∗ = 1.14822731. xn 0.0 0.1 0.2 0.3 0.4 yn 2.00000000 1.65620000 1.41097281 1.24645047 1.14796764 init. cond. RK4 RK4 RK4 ABM 7. Using x0 = 1, y0 = 2, and h = 0.1 we have x1 = x0 + h(x0 + y0 ) = 1 + 0.1(1 + 2) = 1.3 and y1 = y0 + h(x0 − y0 ) = 2 + 0.1(1 − 2) = 1.9 337 CHAPTER 6 REVIEW EXERCISES x2 = x1 + h(x1 + y1 ) = 1.3 + 0.1(1.3 + 1.9) = 1.62 y2 = y1 + h(x1 − y1 ) = 1.9 + 0.1(1.3 − 1.9) = 1.84. Thus, x(0.2) ≈ 1.62 and y(0.2) ≈ 1.84. 8. We identify P (x) = 0, Q(x) = 6.55(1 + x), f (x) = 1, and h = (1 − 0)/10 = 0.1. Then the finite difference equation is yi+1 + [−2 + 0.0655(1 + xi )]yi + yi−1 = 0.001 or yi+1 + (0.0655xi − 1.9345)yi + yi−1 = 0.001. The solution of the corresponding linear system gives x y 0.0 0.0000 0.1 4.1987 0.2 0.3 0.4 0.5 0.6 8.1049 11.3840 13.7038 14.7770 14.4083 0.7 12.5396 338 0.8 9.2847 0.9 4.9450 1.0 0.0000 Part II 7 Vectors, Matrices, and Vector Calculus Vectors EXERCISES 7.1 Vectors in 2-Space √ 1. (a) 6i + 12j (b) i + 8j (c) 3i (d) 2. (a) ⟨3, 3⟩ (b) ⟨3, 4⟩ (c) ⟨−1,−2⟩ (d) 5 3. (a) ⟨12, 0⟩ (b) ⟨4, −5⟩ (c) ⟨4, 5⟩ (d) 4. (a) 1 2i − 12 j (b) 2 3i + 23 j (c) − 31 i − j 5. (a) −9i + 6j (b) −3i + 9j (c) −3i − 5j 6. (a) ⟨3, 9⟩ (b) ⟨−4,−12⟩ (c) ⟨6, 18⟩ 7. (a) −6i + 27j (b) 0 (c) −4i + 18j 8. (a) ⟨21, 30⟩ (b) ⟨8, 12⟩ (c) ⟨6, 8⟩ 9. (a) ⟨4, −12⟩ − ⟨−2, 2⟩ = ⟨6, −14⟩ 65 (e) 3 (e) √ 5 (e) √ 41 √ (d) 2 2/3 (e) √ 10/3 √ (d) 3 10 (e) √ 34 √ (d) 4 10 √ (e) 6 10 (d) 0 √ (e) 2 85 √ (d) 4 13 (e) 10 √ 41 (b) ⟨−3, 9⟩ − ⟨−5, 5⟩ = ⟨2, 4⟩ 10. (a) (4i + 4j) − (6i − 4j) = −2i + 8j (b) (−3i − 3j) − (15i − 10j) = −18i + 7j 11. (a) (4i − 4j) − (−6i + 8j) = 10i − 12j (b) (−3i + 3j) − (−15i + 20j) = 12i − 17j 12. (a) ⟨8, 0⟩ − ⟨0, −6⟩ = ⟨8, 6⟩ (b) ⟨−6, 0⟩ − ⟨0, −15⟩ = ⟨−6, 15⟩ 13. (a) ⟨16, 40⟩ − ⟨−4, −12⟩ = ⟨20, 52⟩ (b) ⟨−12, −30⟩ − ⟨−10, −30⟩ = ⟨−2, 0⟩ 14. (a) ⟨8, 12⟩ − ⟨10, 6⟩ = ⟨−2, 6⟩ (b) ⟨−6, −9⟩ − ⟨25, 15⟩ = ⟨−31, −24⟩ 339 7.1 Vectors in 2-Space 15. 16. −−−→ P1 P2 = ⟨2, 5⟩ −−−→ P1 P2 = ⟨6, −4⟩ 17. 18. −−−→ P1 P2 = ⟨2, 2⟩ −−−→ P1 P2 = ⟨2, −3⟩ −−−→ −−→ −−→ −−→ −−−→ −−→ 19. Since P1 P2 = OP2 − OP1 , OP2 = P1 P2 + OP1 = (4i + 8j) + (−3i + 10j) = i + 18j, and the terminal point is (1, 18). −−−→ −−→ −−→ −−→ −−→ −−−→ 20. Since P1 P2 = OP2 − OP1 , OP1 = OP2 − P1 P2 = ⟨4, 7⟩ − ⟨−5, −1⟩ = ⟨9, 8⟩, and the initial point is (9, 8). 21. a(= −a), b(= − 14 a), c(= 52 a), e(= 2a), and f (= − 21 a) are parallel to a. 22. We want −3b = a, so c = −3(9) = −27. 23. ⟨6, 15⟩ 24. ⟨5, 2⟩ 25. ∥a∥ = 26. ∥a∥ = √ √ √ 4 + 4 = 2 2 ; (a) u = 1 √ ⟨2, 2⟩ 2 2 = ⟨ √12 , √12 ⟩; (b) −u = ⟨− √12 , − √12 ⟩ 9 + 16 = 5; (a) u = 15 ⟨−3, 4⟩ = ⟨− 35 , 45 ⟩; (b) −u = ⟨ 53 , − 45 ⟩ 27. ∥a∥ = 5; (a) u = 51 ⟨0, −5⟩ = ⟨0, −1⟩; (b) −u = ⟨0, 1⟩ √ √ √ √ 28. ∥a∥ = 1 + 3 = 2; (a) u = 12 ⟨1, − 3 ⟩ = ⟨ 12 , − 23 ⟩; (b) −u = ⟨− 12 , 23 ⟩ √ 1 5 29. ∥a + b∥ = ∥⟨5, 12⟩∥ = 25 + 144 = 13; u = 13 ⟨5, 12⟩ = ⟨ 13 , 12 13 ⟩ √ √ 30. ∥2a − 3b∥ = ∥⟨−5, 4⟩∥ = 25 + 16 = 41 ; u = √141 ⟨−5, 4⟩ = ⟨− √541 , √441 ⟩ 31. ∥a∥ = 32. ∥a∥ = √ 9 + 49 = ! 1 4 + 1 4 = √ 58 ; b = 2( √158 )(3i + 7j) = √1 2 ; b = 3( 1/1√2 )( 12 i − 12 j) = √6 i 58 √ 3 2 2 i + − √14 j 58 √ 3 2 2 j 33. − 34 a = ⟨−3, −15/2⟩ 34. 5(a + b) = 5⟨0, 1⟩ = ⟨0, 5⟩ 35. 36. 37. x = −(a + b) = −a − b 38. x = 2(a − b) = 2a − 2b 340 7.1 39. Vectors in 2-Space 40. b = (−c) − a; (b + c) + a = 0; a + b + c = 0 From Problem 39, e + c + d = 0. But b = e − a and e = a + b, so (a + b) + c + d = 0. 41. From 2i + 3j = k1 b + k2 c = k1 (i + j) + k2 (i − j) = (k1 + k2 )i + (k1 − k2 )j we obtain the system of equations k1 + k2 = 2, k1 − k2 = 3. Solving, we find k1 = 25 and k2 = − 12 . Then a = 52 b − 21 c. 42. From 2i + 3j = k1 b + k2 c = k1 (−2i + 4j) + k2 (5i + 7j) = (−2k1 + 5k2 )i + (4k1 + 7k2 )j we obtain the system of 1 7 equations −2k1 + 5k2 = 2, 4k1 + 7k2 = 3. Solving, we find k1 = 34 and k2 = 17 . 43. From y ′ = 12 x we see that the slope of the tangent line at (2, 2) is 1. A vector with slope 1 is i + j. A unit vector √ is (i + j)/∥i + j∥ = (i + j)/ 2 = √12 i + √12 j. Another unit vector tangent to the curve is − √12 i − √12 j. 44. From y ′ = −2x + 3 we see that the slope of the tangent line at (0, 0) is 3. A vector with slope 3 is i + 3j. A √ unit vector is (i + 3j)/∥i + 3j∥ = (i + 3j)/ 10 = √110 i + √110 j. Another unit vector is − √110 i − √110 j. 45. (a) Since Ff = −Fg , ∥Fg ∥ = ∥Ff ∥ = µ∥Fn ∥ and tan θ = ∥Fg ∥/∥Fn ∥ = µ∥Fn ∥/∥Fn ∥ = µ. (b) θ = tan−1 0.6 ≈ 31◦ 46. Since w + F1 + F2 = 0, −200j + ∥F1 ∥ cos 20◦ i + ∥F1 ∥ sin 20◦ j − ∥F2 ∥ cos 15◦ i + ∥F2 ∥ sin 15◦ j = 0 or (∥F1 ∥ cos 20◦ − ∥F2 ∥ cos 15◦ )i + (∥F1 ∥ sin 20◦ + ∥F2 ∥ sin 15◦ − 200)j = 0. Thus, ∥F1 ∥ cos 20◦ − ∥F2 ∥ cos 15◦ = 0; ∥F1 ∥ sin 20◦ + ∥F2 ∥ sin 15◦ − 200 = 0. Solving this system for ∥F1 ∥ and ∥F2 ∥, we obtain ∥F1 ∥ = 200 cos 15◦ 200 cos 15◦ 200 cos 15◦ = ≈ 336.8 lb = sin 15◦ cos 20◦ + cos 15◦ sin 20◦ sin(15◦ + 20◦ ) sin 35◦ and ∥F2 ∥ = sin 15◦ 200 cos 20◦ 200 cos 20◦ = ≈ 327.7 lb. ◦ ◦ ◦ cos 20 + cos 15 sin 20 sin 35◦ 47. Since y/2a(L2 + y 2 )3/2 is an odd function on [−a, a], Fy = 0. Now, using the fact that L/(L2 + y 2 )3/2 is an even function, we have " " a L a dy L dy y = L tan θ, dy = L sec2 θ dθ = 2 2 3/2 2 a 0 (L + y 2 )3/2 −a 2a(L + y ) −1 " " tan−1 a/L L tan a/L 1 L sec2 θ dθ sec2 θ dθ = = a 0 La 0 sec3 θ L3 (1 + tan2 θ)3/2 −1 # " tan−1 a/L #tan a/L 1 1 cos θ dθ = sin θ ## = La 0 La 0 1 1 a √ = = √ . La L2 + a2 L L2 + a2 √ √ Then Fx = qQ/4πϵ0 L L2 + a2 and F = (qQ/4πϵ0 L L2 + a2 )i. 48. Place one corner of the parallelogram at the origin and let two adja−−→ −−→ cent sides be OP1 and OP2 . Let M be the midpoint of the diagonal connecting P1 and P2 and N be the midpoint of the other diagonal. −−→ −−→ −−→ −−→ −−→ Then OM = 21 (OP1 + OP2 ). Since OP1 + OP2 is the main diagonal of the parallelogram and N is its midpoint, −−→ 1 −−→ −−→ −−→ −−→ ON = 2 (OP1 + OP2 ). Thus, OM = ON and the diagonals bisect each other. 341 7.1 Vectors in 2-Space −−→ −−→ −→ −−→ −−→ −−→ −→ 49. By Problem 39, AB + BC + CA = 0 and AD + DE + EC + CA = 0. From the first equation, −−→ −−→ −−→ −−→ −−→ −→ −−→ AB + BC = −CA. Since D and E are midpoints, AD = 21 AB and EC = 21 BC. Then, −→ −−→ 1 −−→ −→ 1− 2 AB + DE + 2 BC + CA = 0 and 1 −→ −→ 1 −−→ −−→ −−→ −→ 1 −→ DE = −CA − (AB + BC) = −CA − (−CA) = − CA. 2 2 2 Thus, the line segment joining the midpoints D and E is parallel to the side AC and half its length. −−→ −→ −−→ 50. We have OA = 150 cos 20◦ i + 150 sin 20◦ j, AB = 200 cos 113◦ i + 200 sin 113◦ j, BC = 240 cos 190◦ i + 240 sin 190◦ j. Then r = (150 cos 20◦ + 200 cos 113◦ + 240 cos 190◦ )i + (150 sin 20◦ + 200 sin 113◦ + 240 sin 190◦ )j ≈ −173.55i + 193.73j and ∥r∥ ≈ 260.09 miles. EXERCISES 7.2 Vectors in 3-Space 1. – 6. 7. A plane perpendicular to the z-axis, 5 units above the xy-plane 8. A plane perpendicular to the x-axis, 1 unit in front of the yz-plane 9. A line perpendicular to the xy-plane at (2, 3, 0) 10. A single point located at (4, −1, 7) 11. (2, 0, 0), (2, 5, 0), (2, 0, 8), (2, 5, 8), (0, 5, 0), (0, 5, 8), (0, 0, 8), (0, 0, 0) 12. 13. (a) xy-plane: (−2, 5, 0), xz-plane: (−2, 0, 4), yz-plane: (0, 5, 4); (b) (−2, 5, −2) 342 7.2 Vectors in 3-Space (c) Since the shortest distance between a point and a plane is a perpendicular line, the point in the plane x = 3 is (3, 5, 4). 14. We find planes that are parallel to coordinate planes: (a) z = −5; (b) x = 1 and y = −1; (c) z = 2 15. The union of the planes x = 0, y = 0, and z = 0 16. The origin (0, 0, 0) 17. The point (−1, 2, −3) 18. The union of the planes x = 2 and z = 8 19. The union of the planes z = 5 and z = −5 20. The line through the points (1, 1, 1), (−1, −1, −1), and the origin $ √ 21. d = (3 − 6)2 + (−1 − 4)2 + (2 − 8)2 = 70 $ √ 22. d = (−1 − 0)2 + (−3 − 4)2 + (5 − 3)2 = 3 6 $ 23. (a) 7; (b) d = (−3)2 + (−4)2 = 5 $ 24. (a) 2; (b) d = (−6)2 + 22 + (−3)2 = 7 $ √ 25. d(P1 , P2 ) = 32 + 62 + (−6)2 = 9; d(P1 , P3 ) = 22 + 12 + 22 = 3 $ √ d(P2 , P3 ) = (2 − 3)2 + (1 − 6)2 + (2 − (−6))2 = 90 ; The triangle is a right triangle. ! √ √ √ √ 26. d(P1 , P2 ) = 12 + 22 + 42 = 21 ; d(P1 , P3 ) = 32 + 22 + (2 2)2 = 21 ! $ √ √ d(P2 , P3 ) = (3 − 1)2 + (2 − 2)2 + (2 2 − 4)2 = 28 − 16 2 The triangle is an isosceles triangle. $ √ 27. d(P1 , P2 ) = (4 − 1)2 + (1 − 2)2 + (3 − 3)2 = 10 $ √ d(P1 , P3 ) = (4 − 1)2 + (6 − 2)2 + (4 − 3)2 = 26 $ √ d(P2 , P3 ) = (4 − 4)2 + (6 − 1)2 + (4 − 3)2 = 26 ; The triangle is an isosceles triangle. $ 28. d(P1 , P2 ) = (1 − 1)2 + (1 − 1)2 + (1 − (−1))2 = 2 $ d(P1 , P3 ) = (0 − 1)2 + (−1 − 1)2 + (1 − (−1))2 = 3 $ √ d(P2 , P3 ) = (0 − 1)2 + (−1 − 1)2 + (1 − 1)2 = 5 ; The triangle is a right triangle. $ √ 29. d(P1 , P2 ) = (−2 − 1)2 + (−2 − 2)2 + (−3 − 0)2 = 34 $ √ d(P1 , P3 ) = (7 − 1)2 + (10 − 2)2 + (6 − 0)2 = 2 34 $ √ d(P2 , P3 ) = (7 − (−2))2 + (10 − (−2))2 + (6 − (−3))2 = 3 34 Since d(P1 , P2 ) + d(P1 , P3 ) = d(P2 , P3 ), the points P1 , P2 , and P3 are collinear. $ √ 30. d(P1 , P2 ) = (1 − 2)2 + (4 − 3)2 + (4 − 2)2 = 6 $ √ d(P1 , P3 ) = (5 − 2)2 + (0 − 3)2 + (−4 − 2)2 = 3 6 $ √ d(P2 , P3 ) = (5 − 1)2 + (0 − 4)2 + (−4 − 4)2 = 4 6 Since d(P1 , P2 ) + d(P1 , P3 ) = d(P2 , P3 ), the points P1 , P2 , and P3 are collinear. $ √ (2 − x)2 + (1 − 2)2 + (1 − 3)2 = 21 =⇒ x2 − 4x + 9 = 21 =⇒ x2 − 4x + 4 = 16 31. =⇒ (x − 2)2 = 16 =⇒ x = 2 ± 4 or x = 6, −2 343 7.2 Vectors in 3-Space $ (0 − x)2 + (3 − x)2 + (5 − 1)2 = 5 =⇒ 2x2 − 6x + 25 = 25 =⇒ x2 − 3x = 0 =⇒ x = 0, 3 & % 1 + 7 3 + (−2) 1/2 + 5/2 = (4, 1/2, 3/2) , , 33. 2 2 2 % & 0 + 4 5 + 1 −8 + (−6) 34. , , = (2, 3, −7) 2 2 2 32. 35. (x1 + 2)/2 = −1, x1 = −4; (y1 + 3)/2 = −4, y1 = −11; (z1 + 6)/2 = 8, z1 = 10 The coordinates of P1 are (−4, −11, 10). 36. (−3 + (−5))/2 = x3 = −4; (4 + 8)/2 = y3 = 6; (1 + 3)/2 = z3 = 2. The coordinates of P3 are (−4, 6, 2). % & −3 + (−4) 4 + 6 1 + 2 (a) , , = (−7/2, 5, 3/2) 2 2 2 & % −4 + (−5) 6 + 8 2 + 3 (b) , , = (−9/2, 7, 5/2) 2 2 2 −−−→ 37. P1 P2 = ⟨−3, −6, 1⟩ −−−→ 38. P1 P2 = ⟨8, −5/2, 8⟩ −−−→ 39. P1 P2 = ⟨2, 1, 1⟩ −−−→ 40. P1 P2 ⟨−3, −3, 7⟩ 41. a + (b + c) = ⟨2, 4, 12⟩ 42. 2a − (b − c) = ⟨2, −6, 4⟩ − ⟨−3, −5, −8⟩ = ⟨5, −1, 12⟩ 43. b + 2(a − 3c) = ⟨−1, 1, 1⟩ + 2⟨−5, −21, −25⟩ = ⟨−11, −41, −49⟩ 44. 4(a + 2c) − 6b = 4⟨5, 9, 20⟩ − ⟨−6, 6, 6⟩ = ⟨26, 30, 74⟩ √ √ 45. ∥a + c∥ = ∥⟨3, 3, 11⟩∥ = 9 + 9 + 121 = 139 √ √ √ 46. ∥c∥∥2b∥ = ( 4 + 36 + 81 )(2)( 1 + 1 + 1 ) = 22 3 ' ' ' ' ' a ' ' b ' ' ' ' = 1 ∥a∥ + 5 1 ∥b∥ = 1 + 5 = 6 ' 47. ' + 5' ' ∥a∥ |b∥ ' ∥a∥ ∥b∥ √ √ √ √ √ √ √ √ 48. ∥b∥a + ∥a∥b = 1 + 1 + 1 ⟨1, −3, 2⟩ + 1 + 9 + 4 ⟨−1, 1, 1⟩ = ⟨ 3 , −3 3 , 2 3 ⟩ + ⟨− 14 , 14 , 14 ⟩ √ √ √ √ √ √ = ⟨ 3 − 14 , −3 3 + 14 , 2 3 + 14 ⟩ 49. ∥a∥ = 50. ∥a∥ = √ √ 100 + 25 + 100 = 15; u = − 1+9+4= √ 1 ⟨10, −5, 10⟩ = ⟨−2/3, 1/3, −2/3⟩ 15 1 1 3 2 14 ; u = √ (i − 3j + 2k) = √ i − √ j + √ k 14 14 14 14 51. b = 4a = 4i − 4j + 4k % & √ 3 1 3 1 1 ⟨−6, 3, −2⟩ = ⟨ , − , ⟩ 52. ∥a∥ = 36 + 9 + 4 = 7; b = − 2 7 7 14 7 53. 344 7.3 Dot Product EXERCISES 7.3 Dot Product √ 1. a · b = 10(5) cos(π/4) = 25 2 √ 2. a · b = 6(12) cos(π/6) = 36 3 3. a · b = 2(−1) + (−3)2 + 4(5) = 12 4. b · c = (−1)3 + 2(6) + 5(−1) = 4 5. a · c = 2(3) + (−3)6 + 4(−1) = −16 6. a · (b + c) = 2(2) + (−3)8 + 4(4) = −4 7. a · (4b) = 2(−4) + (−3)8 + 4(20) = 48 8. b · (a − c) = (−1)(−1) + 2(−9) + 5(5) = 8 9. a · a = 22 + (−3)2 + 42 = 29 10. (2b) · (3c) = (−2)9 + 4(18) + 10(−3) = 24 11. a · (a + b + c) = 2(4) + (−3)5 + 4(8) = 25 12. (2a) · (a − 2b) = 4(4) + (−6)(−7) + 8(−6) = 10 % ) & ( a·b 12 2(−1) + (−3)2 + 4(5) 13. ⟨−1, 2, 5⟩ = b= ⟨−1, 2, 5⟩ = ⟨−2/5, 4/5, 2⟩ b·b (−1)2 + 22 + 52 30 14. (c · b)a = [3(−1) + 6(2) + (−1)5]⟨2, −3, 4⟩ = 4⟨2, −3, 4⟩ = ⟨8, −12, 16⟩ 15. a and f, b and e, c and d 16. (a) a · b = 2 · 3 + (−c)2 + 3(4) = 0 =⇒ c = 9 (b) a · b = c(−3) + 21 (4) + c2 = c2 − 3c + 2 = (c − 2)(c − 1) = 0 =⇒ c = 1, 2 17. Solving the system of equations 3x1 + y1 − 1 = 0, −3x1 + 2y1 + 2 = 0 gives x1 = 4/9 and y1 = −1/3. Thus, v = ⟨4/9, −1/3, 1⟩. 18. If a and b represent adjacent sides of the rhombus, then ∥a∥ = ∥b∥, the diagonals of the rhombus are a + b and a − b, and (a + b) · (a − b) = a · a − a · b + b · a − b · b = a · a − b · b = ∥a∥2 − ∥b∥2 = 0. Thus, the diagonals are perpendicular. 19. Since c·a= % & a·b a·b a·b b− a ·a=b·a− (a · a) = b · a − ∥a∥2 = b · a − a · b = 0, ∥a∥2 ∥a∥2 ∥a∥2 the vectors c and a are orthogonal. √ √ 20. a · b = 1(1) + c(1) = c + 1; ∥a∥ = 1 + c2 , ∥b∥ = 2 $ c+1 1 √ 1 + c2 = c + 1 =⇒ 1 + c2 = c2 + 2c + 1 =⇒ c = 0 =⇒ cos 45◦ = √ = √ 2 2 1+c 2 √ √ 21. a · b = 3(2) + (−1)2 = 4; ∥a∥ = 10 , ∥b∥ = 2 2 4 1 1 √ =√ cos θ = √ =⇒ θ = cos−1 √ ≈ 1.11 rad ≈ 63.43◦ ( 10)(2 2) 5 5 √ 22. a · b = 2(−3) + 1(−4) = −10; ∥a∥ = 5 , ∥b∥ = 5 345 7.3 Dot Product −10 2 cos θ = √ = −√ ( 5 )5 5 √ =⇒ θ = cos−1 (−2/ 5 ) ≈ 2.68 rad ≈ 153.43◦ √ √ 23. a · b = 2(−1) + 4(−1) + 0(4) = −6; ∥a∥ = 2 5 , ∥b∥ = 3 2 √ −6 1 √ = −√ cos θ = √ =⇒ θ = cos−1 (−1/ 10 ) ≈ 1.89 rad ≈ 108.43◦ (2 5)(3 2) 10 √ √ 24. a · b = 12 (2) + 12 (−4) + 32 (6) = 8; ∥a∥ = 11/2, ∥b∥ = 2 14 √ 8 8 √ =√ =⇒ θ = cos−1 (8/ 154 ) ≈ 0.87 rad ≈ 49.86◦ cos θ = √ ( 11/2)(2 14 ) 154 √ √ √ √ 25. ∥a∥ = 14 ; cos α = 1/ 14 , α ≈ 74.50◦ ; cos β = 2/ 14 , β ≈ 57.69◦ ; cos γ = 3/ 14 , γ ≈ 36.70◦ 26. ∥a∥ = 9; cos α = 2/3, α ≈ 48.19◦ ; cos β = 2/3, β ≈ 48.19◦ ; cos γ = −1/3, γ ≈ 109.47◦ √ 27. ∥a∥ = 2; cos α = 1/2, α = 60◦ ; cos β = 0, β = 90◦ ; cos γ = − 3/2, γ = 150◦ √ √ √ √ 28. ∥a∥ = 78 ; cos α = 5/ 78 , α ≈ 55.52◦ ; cos β = 7/ 78 , β ≈ 37.57◦ ; cos γ = 2/ 78 , γ ≈ 76.91◦ −−→ −−→ −−→ 29. Let θ be the angle between AD and AB and a be the length of an edge of the cube. Then AD = ai + aj + ak, −−→ AB = ai and −−→ −−→ AD · AB 1 a2 √ =√ cos θ = −−→ −−→ = √ 2 2 3 ∥AD∥ ∥AB∥ 3a a −−→ −→ −→ ◦ so θ ≈ 0.955317 radian or 54.7356 . Letting φ be the angle between AD and AC and noting that AC = ai + aj we have * −−→ −→ a2 + a2 AD · AC 2 √ √ cos φ = −−→ −→ = = 2 2 3 ∥AD∥ ∥AC∥ 3a 2a so φ ≈ 0.61548 radian or 35.2644◦ . 30. If a and b are orthogonal, then a · b = 0 and a1 b1 a2 b2 a3 b3 + + ∥a∥ ∥b∥ ∥a∥ ∥b∥ ∥a∥ ∥b∥ 1 1 = (a1 b1 + a2 b2 + a3 b3 ) = (a · b) = 0. ∥a∥ ∥b∥ ∥a∥ ∥b∥ √ √ √ √ 31. a = ⟨5, 7, 4⟩; ∥a∥ = 3 10 ; cos α = 5/3 10 , α ≈ 58.19◦ ; cos β = 7/3 10 , β ≈ 42.45◦ ; cos γ = 4/3 10 , cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 = γ ≈ 65.06◦ 32. We want cos α = cos β = cos γ or a1 = a2 = a3 . Letting a1 = a2 = a3 = 1 we obtain the vector i + j + k. A unit vector in the same direction is √13 i + √13 j + √13 k. 33. compb a = a · b/∥b∥ = ⟨1, −1, 3⟩ · ⟨2, 6, 3⟩/7 = 5/7 √ √ 34. compa b = b · a/∥a∥ = ⟨2, 6, 3⟩ · ⟨1, −1, 3⟩/ 11 = 5/ 11 √ √ 35. b − a = ⟨1, 7, 0⟩; compa (b − a) = (b − a) · a/∥a∥ = ⟨1, 7, 0⟩ · ⟨1, −1, 3⟩/ 11 = −6/ 11 36. a + b = ⟨3, 5, 6⟩; 2b = ⟨4, 12, 6⟩; comp2b (a + b) · 2b/|2b| = ⟨3, 5, 6⟩ · ⟨4, 12, 6⟩/14 = 54/7 √ √ √ −−→ −−→ −−→ −−→ 37. OP = 3i + 10j; ∥OP ∥ = 109 ; comp− → a = a · OP /∥OP ∥ = (4i + 6j) · (3i + 10j)/ 109 = 72/ 109 OP √ √ −−→ −−→ −−→ −−→ 38. OP = ⟨1, −1, 1⟩; ∥OP ∥ = 3 ; comp− → a = a · OP /∥OP ∥ = ⟨2, 1, −1⟩ · ⟨1, −1, 1⟩/ 3 = 0 OP 39. compb a = a · b/∥b∥ = (−5i + 5j) · (−3i + 4j)/5 = 7 28 projb a = (compb a)b/∥b∥ = 7(−3i + 4j)/5 = − 21 5 i+ 5 j √ √ 40. compb a = a · b/∥b∥ = (4i + 2j) · (−3i + j)/ 10 = − 10 √ √ projb a = (compb a)b/∥b∥ = − 10(−3i + j)/ 10 = 3i − j 346 7.3 Dot Product 41. compb a = a · b/∥b∥ = (−i − 2j + 7k) · (6i − 3j − 2k)/7 = −2 6 4 projb a = (compb a)b/∥b∥ = −2(6i − 3j − 2k)/7 = − 12 7 i + 7j + 7k 42. compb a = a · b/∥b∥ = ⟨1, 1, 1⟩ · ⟨−2, 2, −1⟩/3 = −1/3 projb a = (compb a)b/∥b∥ = − 13 ⟨−2, 2, −1⟩/3 = ⟨2/9, −2/9, 1/9⟩ 43. a + b = 3i + 4j; ∥a + b∥ = 5; comp(a+b) a = a · (a + b)/∥a + b∥ = (4i + 3j) · (3i + 4j)/5 = 24/5 72 96 proj(a+b) a = (comp(a+b) a)(a + b)/∥a + b∥ = 24 5 (3i + 4j)/5 = 25 i + 25 j √ √ √ 44. a − b = 5i + 2j; ∥a − b∥ = 29 ; comp(a−b) b = b · (a − b)/∥a − b∥ = (−i + j) · (5i + 2j)/ 29 = −3/ 29 √ 6 15 i − 29 j proj(a−b) b = (comp(a−b) b)(a − b)/∥a − b∥ = − √329 (5i + 2j)/ 29 = − 29 45. We identify ∥F∥ = 20, θ = 60◦ and ∥d∥ = 100. Then W = ∥F∥ ∥d∥ cos θ = 20(100)( 12 ) = 1000 ft-lb. 46. We identify d = −i + 3j + 8k. Then W = F · d = ⟨4, 3, 5⟩ · ⟨−1, 3, 8⟩ = 45 N-m. 47. (a) Since w and d are orthogonal, W = w · d = 0. √ (b) We identify θ = 0◦ . Then W = ∥F∥ ∥d∥ cos θ = 30( 42 + 32 ) = 150 N-m. 48. Using d = 6i + 2j and F = 3( 53 i + 45 j), W = F · d = ⟨ 59 , 12 5 ⟩ · ⟨6, 2⟩ = 78 5 ft-lb. 49. Let a and b be vectors from the center of the carbon atom to the centers of two distinct hydrogen atoms. The distance between two hydrogen atoms is then $ √ (b − a) · (b − a) = b · b − 2a · b + a · a $ $ = ∥b∥2 + ∥a∥2 − 2∥a∥ ∥b∥ cos θ = (1.1)2 + (1.1)2 − 2(1.1)(1.1) cos 109.5◦ $ = 1.21 + 1.21 − 2.42(−0.333807) ≈ 1.80 angstroms. ∥b − a∥ = 50. Using the fact that | cos θ| ≤ 1, we have |a · b| = ∥a∥ ∥b∥| cos θ| = ∥a∥ ∥b∥| cos θ| ≤ ∥a∥ ∥b∥. 51. ∥a + b∥2 = (a + b) · (a + b) = a · a + 2a · b + b · b = ∥a∥2 + 2a · b + ∥b∥2 ≤ ∥a∥2 + 2|a · b| + ∥b∥2 since x ≤ |x| ≤ ∥a∥2 + 2∥a∥ ∥b∥ + ∥b∥2 = (∥a∥ + ∥b∥)2 by Problem 50 Thus, since ∥a + b∥ and ∥a∥ + ∥b∥ are positive, ∥a + b∥ ≤ ∥a∥ + ∥b∥. 52. Let P1 (x1 , y1 ) and P2 (x2 , y2 ) be distinct points on the line ax + by = −c. Then −−−→ n · P1 P2 = ⟨a, b⟩ · ⟨x2 − x1 , y2 − y1 ⟩ = ax2 − ax1 + by2 − by1 = (ax2 + by2 ) − (ax1 + by1 ) = −c − (−c) = 0, and the vectors are perpendicular. Thus, n is perpendicular to the line. −−−→ 53. Let θ be the angle between n and P2 P1 . Then −−−→ |ax1 − ax2 + by1 − by2 | |⟨a, b⟩ · ⟨x1 − x2 , y1 − y2 ⟩| |n · P2 P1 | −−−→ √ √ = d = ∥P1 P2 ∥ | cos θ| = = ∥n∥ a2 + b2 a2 + b2 |ax1 + by1 − (ax2 + by2 )| |ax1 + by1 − (−c)| |ax1 + by1 + c| √ √ √ = = = . 2 2 2 2 a +b a +b a2 + b2 347 7.4 Product 7.3 Cross Dot Product EXERCISES 7.4 Cross Product 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. # # #i j k ## # # # # # # # # 1 −1 # #1 0# # ## −1 0 ## # # k = −5i − 5j + 3k # # # j+# i−# a × b = # 1 −1 0 # = # # # 0 3 # 0 5# 3 5# #0 3 5 # # # #i j k # # # # # # # # 1 0 ## # # # # # # # # i − # 2 0 # j + # 2 1 # k = −i + 2j − 4k a × b = # 2 1 0 # = ## # # # # # # 0 −1 4 −1 4 0# # 4 0 −1 # # # #i j k ## # # # # # # # # 1 −3 # #1 1# # ## −3 1 ## # # k = ⟨−12, −2, 6⟩ # # # j+# i−# a × b = # 1 −3 1 # = # # # 2 0 # 2 4# 0 4# #2 0 4 # # # # i j k ## # # # # # # # # 1 1# # 1 1# # ## 1 1 ## # # # # # a×b=# 1 1 1#=# # i − # −5 3 # j + # −5 2 # k = ⟨1, −8, 7⟩ # # 2 3 # −5 2 3 # # # # i j k ## # # # # # # # # 2 −1 # # 2 2 ## # # −1 2 ## # # # # j + i − a × b = # 2 −1 2 # = ## # −1 3 # k = −5i + 5k # −1 −1 # # # 3 −1 # # −1 3 −1 # # # #i j k # # # # # # # # 1 −5 ## # #4 1# # 4 −5 # # # # #j + # #i − # a × b = # 4 1 −5 # = ## # 2 3 # k = 14i − 6j + 10k # 2 −1 # # # 3 −1 # # 2 3 −1 # # # # i j k ## # # # # # # # # 1/2 1/2 # # 1/2 0 # # # ## 0 1/2 ## # # # # k = ⟨−3, 2, 3⟩ a × b = # 1/2 0 1/2 # = # #i − # 4 #j + # 4 # # # 6 0 0 6 # 4 6 0 # # # #i j k ## # # # # # # # #0 5 # #0 0# # ## 5 0 ## # # k = ⟨20, 0, −10⟩ # # # j+# i−# a × b = #0 5 0 # = # # # 2 −3 # 2 4# −3 4 # # 2 −3 4 # # # # i j k ## # # # # # # # # 2 # 2 −4 # 2 ## # # 2 −4 ## # #j + # # i − 2 −4 # = ## a×b=# 2 # −3 −3 # k = ⟨0, 0, 0⟩ # −3 6 # # # −3 6 # # −3 −3 6 # # # #i j k ## # # # # # # # #8 1 # # 8 −6 # # ## 1 −6 ## # # k = ⟨−2, −86, −17⟩ # # # j+# i−# a × b = # 8 1 −6 # = # # # 1 −2 # 1 10 # −2 10 # # 1 −2 10 # −−−→ −−−→ 11. P1 P2 = (−2, 2, −4); P1 P3 = (−3, 1, 1) # # # i j k ## # # # # # −2 −−−→ −−−→ # # ## 2 −4 ## i − ## P1 P2 × P1 P3 = # −2 2 −4 # = # # # # −3 1 1 # −3 1 1 # # # # −2 −4 ## j + ## # −3 1 348 # 2 ## k = 6i + 14j + 4k 1# 7.4 # # # i j k# # # # # 1 1 ## # # −−−→ −−−→ −−−→ −−−→ # # #i − #0 12. P1 P2 = (0, 1, 1); P1 P3 = (1, 2, 2); P1 P2 × P1 P3 = # 0 1 1 # = ## # #1 # # 2 2 #1 2 2 # # # #i j k # # # # # # # # # 7 −4 ## #2 7# # 2 −4 # # # # # # # # # 13. a × b = # 2 7 −4 # = # # i − # 1 −1 # j + # 1 1 # k = −3i − 2j − 5k # # 1 −1 # 1 1 −1 # is perpendicular to # # i j # # 14. a × b = # −1 −2 # # 4 −1 both a and b. # k ## # # # # −1 # ## −2 4 ## 4#=# i − ## # # −1 0 4 0# is perpendicular to both a and b. # # #i j k ## # # # # #5 # # # −2 1 ## # 15. a × b = # 5 −2 1 # = ## i − # #2 # # 0 −7 # 2 0 −7 # # # # 1# #j + #0 # #1 2 Cross Product # 1# #k = j − k 2# # # # # # 4# # j + # −1 −2 # k = ⟨4, 16, 9⟩ # # 0 4 −1 # # # # # # 1 # # j + # 5 −2 # k = ⟨14, 37, 4⟩ # # −7 2 0 # a · (a × b) = ⟨5, −2, −1⟩ · ⟨14, 37, 4⟩ = 70 − 74 + 4 = 0; b · (a × b) = ⟨2, 0, −7⟩ · ⟨14, 37, 4⟩ = 28 + 0 − 28 = 0 # # # i j k ## # # # # # # # # 1/2 0 # # # # # # −1/4 0 ## # # j + # 1/2 −1/4 # k = − 3 i − 3j − 1 k 16. a × b = # 1/2 −1/4 0 # = ## i − 2 2 # # # # # # −2 6 2 6 2 −2 # # 2 −2 6# a · (a × b) = ( 21 i − 14 j) · (− 32 i − 3j − 21 k) = − 34 + (− 23 i b · (a × b) = (2i − 2j + 6k) · − 3j − # # # i j k# # # # # 1 1 ## # #2 # # # # i − ## 17. (a) b × c = # 2 1 1 # = # # # # 3 1 1 #3 1 1 # # # #i j k ## # # # # −1 # a × (b × c) = # 1 −1 2 # = ## # # 1 # 0 1 −1 # 1 2 k) 3 4 +0=0 = −3 + 6 − 3 = 0 # # #2 1 ## j + ## # 3 1 # 1 ## k=j−k 1# # # # # # # 1 −1 # #1 2 # 2 ## # k = −i + j + k # # # j+# i−# 0 1 # 0 −1 # −1 # (b) a · c = (i − j + 2k) · (3i + j + k) = 4; (a · c)b = 4(2i + j + k) = 8i + 4j + 4k a · b = (i − j + 2k) · (2i + j + k) = 3; (a · b)c = 3(3i + j + k) = 9i + 3j + 3k a × (b × c) = (a · c)b − (a · b)c = (8i + 4j + 4k) − (9i + 3j + 3k) = −i + j + k # # # i j k ## # # # # # # # # 1 2# # 1 −1 # # ## 2 −1 ## # # k = 21i − 7j + 7k # # # j+# i−# 18. (a) b × c = # 1 2 −1 # = # # # −1 5 # −1 8 # 5 8 # # −1 5 8 # # # # i j k ## # # # # # # # # 3 # 3 −4 # 0 ## # # 0 −4 ## # #j + # # i − 0 −4 # = ## a × (b × c) = # 3 # 21 −7 # k = −28i − 105j − 21k # 21 7 # # # −7 7 # # 21 −7 7 # (b) a · c = (3i − 4k) · (−i + 5j + 8k) = −35; (a · c)b = −35(i + 2j − k) = −35i − 70j + 35k a · b = (3i − 4k) · (i + 2j − k) = 7; (a · b)c = 7(−i + 5j + 8k) = −7i + 35j + 56k a × (b × c) = (a · c)b − (a · b)c = (−35i − 70j + 35k) − (−7i + 35j + 56k) = −28i − 105j − 21k 19. (2i) × j = 2(i × j) = 2k 20. i × (−3k) = −3(i × k) = −3(−j) = 3j 349 7.4 Cross Product 21. k × (2i − j) = k × (2i) + k × (−j) = 2(k × i) − (k × j) = 2j − (−i) = i + 2j 22. i × (j × k) = i × i = 0 23. [(2k) × (3j)] × (4j) = [2 · 3(k × j) × (4j)] = 6(−i) × 4j = (−6)(4)(i × j) = −24k 24. (2i − j + 5k) × i = (2i × i) + (−j × i) + (5k × i) = 2(i × i) + (i × j) + 5(k × i) = 5j + k 25. (i + j) × (i + 5k) = [(i + j) × i] + [(i + j) × 5k] = (i × i) + (j × i) + (i × 5k) + (j × 5k) = −k + 5(−j) + 5i = 5i − 5j − k 26. i × k − 2(j × i) = −j − 2(−k) = −j + 2k 27. k · (j × k) = k · i = 0 28. i · [j × (−k)] = i · [−(j × k)] = i · (−i) = −(i · i) = −1 √ 29. ∥4j − 5(i × j)∥ = ∥4j − 5k∥ = 41 30. (i × j) · (3j × i) = k · (−3k) = −3(k · k) = −3 31. i × (i × j) = i × k = −j 32. (i × j) × i = k × i = j 33. (i × i) × j = 0 × j = 0 34. (i · i)(i × j) = 1(k) = k 35. 2j · [i × (j − 3k)] = 2j · [(i × j) + (i × (−3k)] = 2j · [k + 3(k × i)] = 2j · (k + 3j) = 2j · k + 2j · 3j = 2(j · k) + 6(j · j) = 2(0) + 6(1) = 6 36. (i × k) × (j × i) = (−j) × (−k) = (−1)(−1)(j × k) = j × k = i 37. a × (3b) = 3(a × b) = 3(4i − 3j + 6k) = 12i − 9j + 18k 38. b × a = −a × b = −(a × b) = −4i + 3j − 6k 39. (−a) × b = −(a × b) = −4i + 3j − 6k $ √ 40. |a × b| = 42 + (−3)2 + 62 = 61 # # #i j k ## # # # # # # # # 4 −3 # #4 6 # # # # −3 6 ## # #j + # # 41. (a × b) × c = # 4 −3 6 # = ## i − # 2 −4 # k = −21i + 16j + 22k # # 2 −1 # # # 4 −1 # 2 4 −1 # 42. (a × b) · c = 4(2) + (−3)4 + 6(−1) = −10 43. a · (b × c) = (a × b) · c = 4(2) + (−3)4 + 6(−1) = −10 44. (4a) · (b × c) = (4a × b) · c = 4(a × b) · c = 16(2) + (−12)4 + 24(−1) = −40 −−→ −→ 45. (a) Let A = (1, 3, 0), B = (2, 0, 0), C = (0, 0, 4), and D = (1, −3, 4). Then AB = i − 3j, AC = −i − 3j + 4k, −→ −−→ −−→ −−→ −−→ −−→ CD = i − 3j, and BD = −i − 3j + 4k. Since AB = CD and AC = BD, the quadrilateral is a parallelogram. (b) Computing # # # i j k ## # −−→ −→ # # AB × AC = # 1 −3 0 # = −12i − 4j − 6k # # # −1 −3 4 # √ we find that the area is ∥ − 12i − 4j − 6k∥ = 144 + 16 + 36 = 14. −−→ −→ 46. (a) Let A = (3, 4, 1), B = (−1, 4, 2), C = (2, 0, 2) and D = (−2, 0, 3). Then AB = −4i + k, AC = −i − 4j + k, −−→ −−→ −−→ −−→ −→ −−→ CD = −4i+k, and BD = −i−4j+k. Since AB = CD and AC = BD, the quadrilateral is a parallelogram. (b) Computing # # i # −−→ −→ # AB × AC = # −4 # # −1 # k ## # 1 # = 4i + 3j + 16k # −4 1 # j 0 350 7.4 √ √ we find that the area is ∥4i + 3j + 16k∥ = 16 + 9 + 256 = 281 ≈ 16.76. −−−→ −−−→ 47. P1 P2 = j; P2 P3 = −j + k # # #i j k ## # # # # # # # #0 0# #0 1 # −−−→ −−−→ # # # 1 0 ## 1 # # #j + # P1 P2 × P2 P3 = # 0 1 0 # = ## i − # 0 −1 # k = i; A = 2 ∥i∥ = # #0 1# # # −1 1 # 0 −1 1 # −−−→ −−−→ 48. P1 P2 = j + 2k; P2 P3 = 2i + j − 2k # # #i j k # # # # # # # # # 1 2 ## #0 1# #0 2 # −−−→ −−−→ # # # #j + # #i − # P1 P2 × P2 P3 = # 0 1 2 # = ## # 2 1 # k = −4i + 4j − 2k # 2 −2 # # # 1 −2 # # 2 1 −2 # 1 2 Cross Product sq. unit A = 21 ∥ − 4i + 4j − 2k∥ = 3 sq. units −−−→ −−−→ 49. P1 P2 = −3j − k; P2 P3 = −2i − k # # # i j k ## # # # # # # # # 0 −3 # # 0 −1 # −−−→ −−−→ # # # −3 −1 ## # #j + # # i − P1 P2 × P2 P3 = # 0 −3 −1 # = ## # −2 0 # k = 3i + 2j − 6k # −2 −1 # # # 0 −1 # # −2 0 −1 # A = 12 ∥3i + 2j − 6k∥ = 72 sq. units −−−→ −−−→ 50. P1 P2 = −i + 3k; P2 P3 = 2i + 4j − k # # # i j k ## # # # # # −1 −−−→ −−−→ # # # 0 3 ## # i − P1 P2 × P2 P3 = # −1 0 3 # = ## # # 2 # # 4 −1 # 2 4 −1 # √ # # # 3 # # j + # −1 # # 2 −1 # 0# # k = −12i + 5j − 4k 4# A = 21 ∥ − 12i + 5j − 4k∥ = 185 sq. units 2 # # # i j k ## # # # # # # # # −1 4 # # −1 0 # # ## 4 0 ## # # k = 8i + 2j − 10k # # # j+# i−# 51. b × c = # −1 4 0 # = # # # 2 2# 2 2# 2 2# # 2 2 2# v = |a · (b × c)| = |(i + j) · (8i + 2j − 10k)| = |8 + 2 + 0| = 10 cu. units # # # i j k# # # # # # # # 4 1 ## # #1 1# #1 4# # # # # # # # # 52. b × c = # 1 4 1 # = # # i − # 1 5 # j + # 1 1 # k = 19i − 4j − 3k # # 1 5 #1 1 5 # v = |a · (b × c)| = |(3i + j + k) · (19i − 4j − 3k)| = |57 − 4 − 3| = 50 cu. units # # # i j k ## # # # # # # # # −2 −6 # # # # # # 6 −6 ## # # j + # −2 6 # k = 21i − 14j − 21k 53. b × c = # −2 6 −6 # = ## i − # # # # # # 3 2/2 5/2 1/2 5/2 3 # # 5/2 3 1/2 # a · (b × c) = (4i + 6j) · (21i − 14j − 21k) = 84 − 84 + 0 = 0. The vectors are coplanar. −−−→ −−−→ −−−→ 54. The four points will be coplanar if the three vectors P1 P2 = ⟨3, −1, −1⟩, P2 P3 = ⟨−3, −5, 13⟩, and P3 P4 = ⟨−8, 7, −6⟩ are coplanar. # # # i j k ## # # # # # # # # −3 13 # # −3 −5 # −−−→ −−−→ # # ## −5 13 ## # # # # P2 P3 × P3 P4 = # −3 −5 13 # = # # i − # −8 −6 # j + # −8 7 # k = ⟨−61, −122, −61⟩ # # 7 −6 # −8 7 −6 # −−−→ −−−→ −−−→ P1 P2 · (P2 P3 × P3 P4 ) = ⟨3, −1, −1⟩ · ⟨−61, −122, −61⟩ = −183 + 122 + 61 = 0 The four points are coplanar. 55. (a) Since θ = 90◦ , ∥a × b∥ = ∥a∥ ∥b∥ | sin 90◦ | = 6.4(5) = 32. 351 7.4 Cross Product (b) The direction of a × b is into the fourth quadrant of the xy-plane or to the left of the plane determined by a and b as shown in Figure 7.54 in the text. It makes an angle of 30◦ with the positive x-axis. √ √ (c) We identify n = ( 3 i − j)/2. Then a × b = 32n = 16 3 i − 16j. √ √ √ 56. Using Definition 7.4, a × b = 27 (8) sin 120◦ n = 24 3 ( 3/2)n = 36n. By the right-hand rule, n = j or n = −j. Thus, a × b = 36j or −36j. 57. (a) We note first that a × b = k, b × c = 12 (i − k), c × a = 21 (j − k), a · (b × c) = c · (a × b) = 12 . Then A= 1 2 (i − k) 1 2 = i − k, B= 1 2 (j − k) 1 2 = j − k, and C = k 1 2 1 2 , b · (c × a) = 1 2 , and = 2k. (b) We need to compute A · (B × C). Using formula (10) in the text we have (c × a) × (a × b) [(c × a) · b]a − [(c × a) · a]b = [b · (c × a)][c · (a × b)] [b · (c × a)][c · (a × b)] a since (c × a) · a = 0. = c · (a × b) B×C= Then A · (B × C) = a 1 b×c · = a · (b × c) c · (a × b) c · (a × b) and the volume of the unit cell of the reciprocal latrice is the reciprocal of the volume of the unit cell of the original lattice. # # # # # # # a2 # a1 # a1 a3 # a3 # a2 # # # # # # #k 58. a × (b + c) = # i−# j+# b2 + c2 b3 + c3 # b1 + c1 b3 + c3 # b1 + c1 b2 + c2 # = (a2 b3 − a3 b2 )i + (a2 c3 − a3 c2 )i − [(a1 b3 − a3 b1 )j + (a1 c3 − a3 c1 )j] + (a1 b2 − a2 b1 )k + (a1 c2 − a2 c1 )k = (a2 b3 − a3 b2 )i − (a1 b3 − a3 b1 )j + (a1 b2 − a2 b1 )k + (a2 c3 − a3 c2 )i − (a1 c3 − a3 c1 )j + (a1 c2 − a2 c1 )k =a×b+a×c 59. b × c = (b2 c3 − b3 c2 )i − (b1 c3 − b3 c1 )j + (b1 c2 − b2 c1 )k a × (b × c) = [a2 (b1 c2 − b2 c1 ) + a3 (b1 c3 − b3 c1 )]i − [a1 (b1 c2 − b2 c1 ) − a3 (b2 c3 − b3 c2 )]j + [−a1 (b1 c3 − b3 c1 ) − a2 (b2 c3 − b3 c2 )]k = (a2 b1 c2 − a2 b2 c1 + a3 b1 c3 − a3 b3 c1 )i − (a1 b1 c2 − a1 b2 c1 − a3 b2 c3 + a3 b3 c2 )j − (a1 b1 c3 − a1 b3 c1 + a2 b2 c3 − a2 b3 c2 )k (a · c)b − (a · b)c = (a1 c1 + a2 c2 + a3 c3 )(b1 i + b2 j + b3 k) − (a1 b1 + a2 b2 + a3 b3 )(c1 i + c2 j + c3 k) = (a2 b1 c2 − a2 b2 c1 + a3 b1 c3 − a3 b3 c1 )i − (a1 b1 c2 − a1 b2 c1 − a3 b2 c3 + a3 b3 c2 )j − (a1 b1 c3 − a1 b3 c1 + a2 b2 c3 − a2 b3 c2 )k 60. The statement is false since i × (i × j) = i × k = −j and (i × i) × j = 0 × j = 0. 61. Using equation 9 in the text, # # a1 # # a · (b × c) = # b1 # # c1 a2 b2 c2 # a3 ## # b3 # # c3 # and # # c1 # # (a × b) · c = c · (a × b) = # a1 # # b1 c2 a2 b2 # c3 ## # a3 # . # b3 # Expanding these determinants out we obtain a · (b × c) = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a1 b3 c2 − a2 b1 c3 and c · (a × b) = a2 b3 c1 + a3 b1 c2 + a1 b2 c3 − a2 b1 c3 − a3 b2 c1 − a1 b3 c2 . These are equal so a · (b × c) = (a × b) · c. 352 7.5 Lines and Planes in 3-Space 62. a × (b × c) + b × (c × a) + c × (a × b) = (a · c)b − (a · b)c + (b · a)c − (b · c)a + (c · b)a − (c · a)b 63. Since and = [(a · c)b − (c · a)b] + [(b · a)c − (a · b)c] + [(c · b)a − (b · c)a] = 0 ∥a × b∥2 = (a2 b3 − a3 b2 )2 + (a1 b3 − a3 b1 )2 + (a1 b2 − a2 b1 )2 = a22 b23 − 2a2 b3 a3 b2 + a23 b22 + a21 b23 − 2a1 b3 a3 b1 + a23 b21 + a21 b22 − 2a1 b2 a2 b1 + a22 b21 ∥a∥2 ∥b∥2 − (a · b)2 = (a21 + a22 + a23 )(b21 + b22 + b23 ) − (a1 b1 + a2 b2 + a3 b3 )2 = a21 a22 + a21 b22 + a21 b23 + a22 b21 + a22 b22 + a22 b23 + a23 b21 + a23 b22 + a23 b23 − a21 b21 − a22 b22 − a23 b23 − 2a1 b1 a2 b2 − 2a1 b1 a3 b3 − 2a2 b2 a3 b3 = a21 b22 + a21 b23 + a22 b21 + a22 b23 + a23 b21 + a23 b22 − 2a1 a2 b1 b2 − 2a1 a3 b1 b3 − 2a2 a3 b2 b3 we see that ∥a × b∥2 = ∥a∥2 ∥b∥2 − (a · b)2 . 64. No. For example i × (i + j) = i × j by the distributive law (iii) in the text, and the fact that i × i = 0. But i + j does not equal j. 65. By the distributive law (iii) in the text: (a + b) × (a − b) = (a + b) × a − (a + b) × b = a × a + b × a − a × b − b × b = 2b × a since a × a = 0, b × b = 0, and −a × b = b × a. EXERCISES 7.5 Lines and Planes in 3-Space −−→ −−→ The equation of a line through P1 and P2 in 3-space with r1 = OP1 and r2 = OP2 can be expressed as r = r1 + t(ka) or r = r2 + t(ka) where a = r2 − r1 and k is any non-zero scalar. Thus, the form of the equation of a line is not unique. (See the alternate solution to Problem 1.) 1. a = ⟨1 − 3, 2 − 5, 1 − (−2)⟩ = ⟨−2, −3, 3⟩; ⟨x, y, z⟩ = ⟨1, 2, 1⟩ + t⟨−2, −3, 3⟩ Alternate Solution: a = ⟨3 − 1, 5 − 2, −2 − 1⟩ = ⟨2, 3, −3⟩; ⟨x, y, z⟩ = ⟨3, 5, −2⟩ + t⟨2, 3, −3⟩ 2. a = ⟨0 − (−2), 4 − 6, 5 − 3⟩ = ⟨2, −2, 2⟩; ⟨x, y, z⟩ = ⟨0, 4, 5⟩ + t⟨2, −2, 2⟩ 3. a = ⟨1/2 − (−3/2), −1/2 − 5/2, 1 − (−1/2)⟩ = ⟨2, −3, 3/2⟩; ⟨x, y, z⟩ = ⟨1/2, −1/2, 1⟩ + t⟨2, −3, 3/2⟩ 4. a = ⟨10 − 5, 2 − (−3), −10 − 5⟩ = ⟨5, 5, −15⟩; ⟨x, y, z⟩ = ⟨10, 2, −10⟩ + t⟨5, 5, −15⟩ 5. a = ⟨1 − (−4), 1 − 1, −1 − (−1)⟩ = ⟨5, 0, 0⟩; ⟨x, y, z⟩ = ⟨1, 1, −1⟩ + t⟨5, 0, 0⟩ 6. a = ⟨3 − 5/2, 2 − 1, 1 − (−2)⟩ = ⟨1/2, 1, 3⟩; ⟨x, y, z⟩ = ⟨3, 2, 1⟩ + t⟨1/2, 1, 3⟩ 7. a = ⟨2 − 6, 3 − (−1), 5 − 8⟩ = ⟨−4, 4, −3⟩; x = 2 − 4t, y = 3 + 4t, z = 5 − 3t 8. a = ⟨2 − 0, 0 − 4, 0 − 9⟩ = ⟨2, −4, −9⟩; x = 2 + 2t, y = −4t, z = −9t 9. a = ⟨1 − 3, 0 − (−2), 0 − (−7)⟩ = ⟨−2, 2, 7⟩; x = 1 − 2t, y = 2t, z = 7t 10. a = ⟨0 − (−2), 0 − 4, 5 − 0⟩ = ⟨2, −4, 5⟩; x = 2t, y = −4t, z = 5 + 5t 353 7.5 Lines and Planes in 3-Space 11. a = ⟨4 − (−6), 1/2 − (−1/4), 1/3 − 1/6⟩ = ⟨10, 3/4, 1/6⟩; x = 4 + 10t, y = 1 3 1 1 + t, z = + t 2 4 3 6 12. a = ⟨−3 − 4, 7 − (−8), 9 − (−1)⟩ = ⟨−7, 15, 10⟩; x = −3 − 7t, y = 7 + 15t, z = 9 + 10t 13. a1 = 10 − 1 = 9, a2 = 14 − 4 = 10, a3 = −2 − (−9) = 7; y − 14 z+2 x − 10 = = 9 10 7 14. a1 = 1 − 2/3 = 1/3, a2 = 3 − 0 = 3, a3 = 1/4 − (−1/4) = 1/2; 15. a1 = −7 − 4 = −11, a2 = 2 − 2 = 0, a3 = 5 − 1 = 4; x−1 y−3 z − 1/4 = = 1/3 3 1/2 z−5 x+7 = , y=2 −11 4 16. a1 = 1 − (−5) = 6, a2 = 1 − (−2) = 3, a3 = 2 − (−4) = 6; y−1 z−2 x−1 = = 6 3 6 17. a1 = 5 − 5 = 0, a2 = 10 − 1 = 9, a3 = −2 − (−14) = 12; x = 5, y − 10 z+2 = 9 12 18. a1 = 5/6 − 1/3 = 1/2; a2 = −1/4 − 3/8 = −5/8; a3 = 1/5 − 1/10 = 1/10 y + 1/4 z − 1/5 x − 5/6 = = 1/2 −5/8 1/10 19. parametric: x = 4 + 3t, y = 6 + t/2, z = −7 − 3t/2; symmetric: x−4 y−6 z+7 = = 3 1/2 −3/2 x−1 y−8 = , z = −2 −7 −8 y z x = = 21. parametric: x = 5t, y = 9t, z = 4t; symmetric: 5 9 4 x y+3 z − 10 = = 22. parametric: x = 12t, y = −3 − 5t, z = 10 − 6t; symmetric: 12 −5 −6 20. parametric: x = 1 − 7t, y = 8 − 8t, z = −2; symmetric: 23. Writing the given line in the form x/2 = (y − 1)/(−3) = (z − 5)/6, we see that a direction vector is ⟨2, −3, 6⟩. Parametric equations for the line are x = 6 + 2t, y = 4 − 3t, z = −2 + 6t. 24. A direction vector is ⟨5, 1/3, −2⟩. Symmetric equations for the line are (x−4)/5 = (y +11)/(1/3) = (z +7)/(−2). 25. A direction vector parallel to both the xz- and xy-planes is i = ⟨1, 0, 0⟩. Parametric equations for the line are x = 2 + t, y = −2, z = 15. 26. (a) Since the unit vector j = ⟨0, 1, 0⟩ lies along the y-axis, we have x = 1, y = 2 + t, z = 8. (b) since the unit vector k = ⟨0, 0, 1⟩ is perpendicular to the xy-plane, we have x = 1, y = 2, z = 8 + t. 27. Both lines go through the points (0, 0, 0) and (6, 6, 6). Since two points determine a line, the lines are the same. 28. a and f are parallel since ⟨9, −12, 6⟩ = −3⟨−3, 4, −2⟩. c and d are orthogonal since ⟨2, −3, 4⟩ · ⟨1, 4, 5/2⟩ = 0. 29. In the xy-plane, z = 9 + 3t = 0 and t = −3. Then x = 4 − 2(−3) = 10 and y = 1 + 2(−3) = −5. The point is (10, −5, 0). In the xz-plane, y = 1+2t = 0 and t = −1/2. Then x = 4−2(−1/2) = 5 and z = 9+3(−1/2) = 15/2. The point is (5, 0, 15/2). In the yz-plane, x = 4−2t = 0 and t = 2. Then y = 1+2(2) = 5 and z = 9+3(2) = 15. The point is (0, 5, 15). 30. The parametric equations for the line are x = 1 + 2t, y = −2 + 3t, z = 4 + 2t. In the xy-plane, z = 4 + 2t = 0 and t = −2. Then x = 1 + 2(−2) = −3 and y = −2 + 3(−2) = −8. The point is (−3, −8, 0). In the xz-plane, y = −2 + 3t = 0 and t = 2/3. Then x = 1 + 2(2/3) = 7/3 and z = 4 + 2(2/3) = 16/3. The point is (7/3, 0, 16/3). In the yz-plane, x = 1 + 2t = 0 and t = −1/2. Then y = −2 + 3(−1/2) = −7/2 and z = 4 + 2(−1/2) = 3. The point is (0, −7/2, 3). 354 7.5 Lines and Planes in 3-Space 31. Solving the system 4 + t = 6 + 2s, 5 + t = 11 + 4s, −1 + 2t = −3 + s, or t − 2s = 2, t − 4s = 6, 2t − s = −2 yields s = −2 and t = −2 in all three equations. Thus, the lines intersect at the point x = 4 + (−2) = 2, y = 5 + (−2) = 3, z = −1 + 2(−2) = −5, or (2, 3, −5). 32. Solving the system 1 + t = 2 − s, 2 − t = 1 + s, 3t = 6s, or t + s = 1, t + s = 1, t − 2s = 0 yields s = 1/3 and t = 2/3 in all three equations. Thus, the lines intersect at the point x = 1 + 2/3 = 5/3, y = 2 − 2/3 = 4/3, z = 3(2/3) = 2, or (5/3, 4/3, 2). 33. The system of equations 2 − t = 4 + s, 3 + t = 1 + s, 1 + t = 1 − s, or t + s = −2, t − s = −2, t + s = 0 has no solution since −2 ̸= 0. Thus, the lines do not intersect. 34. Solving the system 3 − t = 2 + 2s, 2 + t = −2 + 3s, 8 + 2t = −2 + 8s, or t + 2s = 1, t − 3s = −4, 2t − 8s = −10 yields s = 1 and t = −1 in all three equations. Thus, the lines intersect at the point x = 3 − (−1) = 4, y = 2 + (−1) = 1, z = 8 + 2(−1) = 6, or (4, 1, 6). 35. a = ⟨−1, 2, −2⟩, b = ⟨2, 3, −6⟩, a · b = 16, ∥a∥ = 3, ∥b∥ = 7; cos θ = a·b 16 = ; ∥a∥ ∥b∥ 3·7 16 ≈ 40.37◦ 21 √ √ 36. a = ⟨2, 7, −1⟩, b = ⟨−2, 1, 4⟩, a · b = −1, ∥a∥ = 3 6 , ∥b∥ = 21 ; 1 a·b 1 −1 = − √ ; θ = cos−1 (− √ ) ≈ 91.70◦ cos θ = = √ √ ∥a∥ ∥b∥ (3 6 )( 21 ) 9 14 9 14 θ = cos−1 37. A direction vector perpendicular to the given lines will be ⟨1, 1, 1⟩ × ⟨−2, 1, −5⟩ = ⟨−6, 3, 3⟩. Equations of the line are x = 4 − 6t, y = 1 + 3t, z = 6 + 3t. 38. The direction vectors of the given lines are ⟨3, 2, 4⟩ and ⟨6, 4, 8⟩ = 2⟨3, 2, 4⟩. These are parallel, so we need a third vector parallel to the plane containing the lines which is not parallel to them. The point (1, −1, 0) is on the first line and (−4, 6, 10) is on the second line. A third vector is then ⟨1, −1, 0⟩ − ⟨−4, 6, 10⟩ = ⟨5, −7, −10⟩. Now a direction vector perpendicular to the plane is ⟨3, 2, 4⟩ × ⟨5, −7, −10⟩ = ⟨8, 50, −31⟩. Equations of the line through (1, −1, 0) and perpendicular to the plane are x = 1 + 8t, y = −1 + 50t, z = −31t. 39. 2(x − 5) − 3(y − 1) + 4(z − 3) = 0; 2x − 3y + 4z = 19 40. 4(x − 1) − 2(y − 2) + 0(z − 5) = 0; 4x − 2y = 0 41. −5(x − 6) + 0(y − 10) + 3(z + 7) = 0; −5x + 3z = −51 42. 6x − y + 3z = 0 43. 6(x − 1/2) + 8(y − 3/4) − 4(z + 1/2) = 0; 6x + 8y − 4z = 11 44. −(x + 1) + (y − 1) − (z − 0) = 0; −x + y − z = 2 45. From the points (3, 5, 2) and (2, 3, 1) we obtain the vector u = i + 2j + k. From the points (2, 3, 1) and (−1, −1, 4) we obtain the vector v = 3i + 4j − 3k. From the points (−1, −1, 4) and (x, y, z) we obtain the vector w = (x + 1)i + (y + 1)j + (z − 4)k. Then, a normal vector is # # #i j k # # # # # u × v = # 1 2 1 # = −10i + 6j − 2k. # # # 3 4 −3 # A vector equation of the plane is −10(x + 1) + 6(y + 1) − 2(z − 4) = 0 or 5x − 3y + z = 2. 46. From the points (0, 1, 0) and (0, 1, 1) we obtain the vector u = k. From the points (0, 1, 1) and (1, 3, −1) we obtain the vector v = i + 2j − 2k. From the points (1, 3, −1) and (x, y, z) we obtain the vector 355 7.5 Lines and Planes in 3-Space w = (x − 1)i + (y − 3)j + (z + 1)k. Then, a normal # #i # # u × v = #0 # #1 vector is # j k ## # 0 1 # = −2i + j. # 2 −2 # A vector equation of the plane is −2(x − 1) + (y − 3) + 0(z + 1) = 0 or −2x + y = 1. 47. From the points (0, 0, 0) and (1, 1, 1) we obtain the vector u = i + j + k. From the points (1, 1, 1) and (3, 2, −1) we obtain the vector v = 2i + j − 2k. From the points (3, 2, −1) and (x, y, z) we obtain the vector w = (x − 3)i + (y − 2)j + (z + 1)k. Then, a normal vector is # # #i j k # # # # # u × v = # 1 1 1 # = −3i + 4j − k. # # # 2 1 −2 # A vector equation of the plane is −3(x − 3) + 4(y − 2) − (z + 1) = 0 or −3x + 4y − z = 0. 48. The three points are not colinear and all satisfy x = 0, which is the equation of the plane. 49. From the points (1, 2, −1) and (4, 3, 1) we obtain the vector u = 3i + j + 2k. From the points (4, 3, 1) and (7, 4, 3) we obtain the vector v = 3i + j + 2k. From the points (7, 4, 3) and (x, y, z) we obtain the vector w = (x − 7)i + (y − 4)j + (z − 3)k. Since u × v = 0, the points are colinear. 50. From the points (2, 1, 2) and (4, 1, 0) we obtain the vector u = 2i − 2k. From the points (4, 1, 0) and (5, 0, −5) we obtain the vector v = i − j − 5k. From the points (5, 0, −5) and (x, y, z) we obtain the vector w = (x − 5)i + yj + (z + 5)k. Then, a normal vector is # # #i j k ## # # # u × v = # 2 0 −2 # = −2i + 8j − 2k. # # # 1 −1 −5 # A vector equation of the plane is −2(x − 5) + 8y − 2(z + 5) = 0 or x − 4y + z = 0. 51. A normal vector to x + y − 4z = 1 is ⟨1, 1, −4⟩. The equation of the parallel plane is (x − 2) + (y − 3) − 4(z + 5) = 0 or x + y − 4z = 25. 52. A normal vector to 5x−y +z = 6 is ⟨5, −1, 1, ⟩. The equation of the parallel plane is 5(x−0)−(y −0)+(z −0) = 0 or 5x − y + z = 0. 53. A normal vector to the xy-plane is ⟨0, 0, 1⟩. The equation of the parallel plane is z − 12 = 0 or z = 12. 54. A normal vector is ⟨0, 1, 0⟩. The equation of the plane is y + 5 = 0 or y = −5. 55. Direction vectors of the lines are ⟨3, −1, 1⟩. and ⟨4, 2, 1⟩. A normal vector to the plane is ⟨3, −1, 1⟩ × ⟨4, 2, 1⟩ = ⟨−3, 1, 10⟩. A point on the first line, and thus in the plane, is ⟨1, 1, 2⟩. The equation of the plane is −3(x − 1) + (y − 1) + 10(z − 2) = 0 or −3x + y + 10z = 18. 56. Direction vectors of the lines are ⟨2, −1, 6⟩ and ⟨1, 1, −3⟩. A normal vector to the plane is ⟨2, −1, 6⟩×⟨1, 1, −3⟩ = ⟨−3, 12, 3⟩. A point on the first line, and thus in the plane, is (1, −1, 5). The equation of the plane is −3(x − 1) + 12(y + 1) + 3(z − 5) = 0 or −x + 4y + z = 0. 57. A direction vector for the two lines is ⟨1, 2, 1⟩. Points on the lines are (1, 1, 3) and (3, 0, −2). Thus, another vector parallel to the plane is ⟨1−3, 1−0, 3+2⟩ = ⟨−2, 1, 5⟩. A normal vector to the plane is ⟨1, 2, 1⟩×⟨−2, 1, 5⟩ = ⟨9, −7, 5⟩. Using the point (3, 0, −2) in the plane, the equation of the plane is 9(x − 3) − 7(y − 0) + 5(z + 2) = 0 or 9x − 7y + 5z = 17. 356 7.5 Lines and Planes in 3-Space 58. A direction vector for the line is ⟨3, 2, −2⟩. Letting t = 0, we see that the origin is on the line and hence in the plane. Thus, another vector parallel to the plane is ⟨4 − 0, 0 − 0, −6 − 0⟩ = ⟨4, 0, −6⟩. A normal vector to the plane is ⟨3, 2, −2⟩ × ⟨4, 0, −6⟩ = ⟨−12, 10, −8⟩. The equation of the plane is −12(x − 0) + 10(y − 0) − 8(z − 0) = 0 or 6x − 5y + 4z = 0. 59. A direction vector for the line, and hence a normal vector to the plane, is ⟨−3, 1, −1/2⟩. The equation of the plane is −3(x − 2) + (y − 4) − 12 (z − 8) = 0 or −3x + y − 12 z = −6. 60. A normal vector to the plane is ⟨2 − 1, 6 − 0, −3 + 2⟩ = ⟨1, 6, −1⟩. The equation of the plane is (x − 1) + 6(y − 1) − (z − 1) = 0 or x + 6y − z = 6. 61. Normal vectors to the planes are (a) ⟨2, −1, 3⟩, (b) ⟨1, 2, 2⟩, (c) ⟨1, 1, −3/2⟩, (d) ⟨−5, 2, 4⟩, (e) ⟨−8, −8, 12⟩, (f ) ⟨−2, 1, −3⟩. Parallel planes are (c) and (e), and (a) and (f ). Perpendicular planes are (a) and (d), (b) and (c), (b) and (e), and (d) and (f ). 62. A normal vector to the plane is ⟨−7, 2, 3⟩. This is a direction vector for the line and the equations of the line are x = −4 − 7t, y = 1 + 2t, z = 7 + 3t. 63. A direction vector of the line is ⟨−6, 9, 3⟩, and the normal vectors of the planes are (a) ⟨4, 1, 2⟩, (b) ⟨2, −3, 1⟩, (c) ⟨10, −15, −5⟩, (d) ⟨−4, 6, 2⟩. Vectors (c) and (d) are multiples of the direction vector and hence the corresponding planes are perpendicular to the line. 64. A direction vector of the line is ⟨−2, 4, 1⟩, and normal vectors to the planes are (a) ⟨1, −1, 3⟩, (b) ⟨6, −3, 0⟩, (c) ⟨1, −2, 5⟩, (d) ⟨−2, 1, −2⟩. Since the dot product of each normal vector with the direction vector is non-zero, none of the planes are parallel to the line. 65. Letting z = t in both equations and solving 5x − 4y = 8 + 9t, x + 4y = 4 − 3t, we obtain x = 2 + t, y = z = t. 1 2 − t, 66. Letting y = t in both equations and solving x − z = 2 − 2t, 3x + 2z = 1 + t, we obtain x = 1 − 53 t, y = t, z = −1 + 57 t or, letting t = 5s, x = 1 − 3s, y = 5s, z = −1 + 7s. 67. Letting z = t in both equations and solving 4x − 2y = 1 + t, x + y = 1 − 2t, we obtain x = z = t. 1 2 − 21 t, y = 1 2 − 23 t, 68. Letting z = t and using y = 0 in the first equation, we obtain x = − 21 t, y = 0, z = t. 69. Substituting the parametric equations into the equation of the plane, we obtain 2(1+2t)−3(2−t)+2(−3t) = −7 or t = −3. Letting t = −3 in the equation of the line, we obtain the point of intersection (−5, 5, 9). 70. Substituting the parametric equations into the equation of the plane, we obtain (3−2t)+(1+6t)+4(2− 12 t) = 12 or 2t = 0. Letting t = 0 in the equation of the line, we obtain the point of intersection (3, 1, 2). 71. Substituting the parametric equations into the equation of the plane, we obtain 1 + 2 − (1 + t) = 8 or t = −6. Letting t = −6 in the equation of the line, we obtain the point of intersection (1, 2, −5). 72. Substituting the parametric equations into the equation of the plane, we obtain 4 + t − 3(2 + t) + 2(1 + 5t) = 0 or t = 0. Letting t = 0 in the equation of the line, we obtain the point of intersection (4, 2, 1). 357 7.5 Lines and Planes in 3-Space In Problems 73 and 74, the cross product of the normal vectors to the two planes will be a vector parallel to both planes, and hence a direction vector for a line parallel to the two planes. 73. Normal vectors are ⟨1, 1, −4⟩ and ⟨2, −1, 1⟩. A direction vector is ⟨1, 1, −4⟩ × ⟨2, −1, 1⟩ = ⟨−3, −9, −3⟩ = −3⟨1, 3, 1⟩. Equations of the line are x = 5 + t, y = 6 + 3t, z = −12 + t. 74. Normal vectors are ⟨2, 0, 1⟩ and ⟨−1, 3, 1⟩. A direction vector is ⟨2, 0, 1⟩ × ⟨−1, 3, 1⟩ = ⟨−3, −3, 6⟩ = −3⟨1, 1, −2⟩. Equations of the line are x = −3 + t, y = 5 + t, z = −1 − 2t. In Problems 75 and 76, the cross product of the direction vector of the line with the normal vector of the given plane will be a normal vector to the desired plane. 75. A direction vector of the line is ⟨3, −1, 5⟩ and a normal vector to the given plane is ⟨1, 1, 1⟩. A normal vector to the desired plane is ⟨3, −1, 5⟩ × ⟨1, 1, 1⟩ = ⟨−6, 2, 4⟩. A point on the line, and hence in the plane, is (4, 0, 1). The equation of the plane is −6(x − 4) + 2(y − 0) + 4(z − 1) = 0 or 3x − y − 2z = 10. 76. A direction vector of the line is ⟨3, 5, 2⟩ and a normal vector to the given plane is ⟨2, −4, −1⟩. A normal vector to the desired plane is ⟨−3, 5, 2⟩ × ⟨2, −4, −1⟩ = ⟨3, 1, 2⟩. A point on the line, and hence in the plane, is (2, −2, 8). The equation of the plane is 3(x − 2) + (y + 2) + 2(z − 8) = 0 or 3x + y + 2z = 20. 77. 78. 79. 80. 81. 82. 358 7.6 Vector Spaces EXERCISES 7.6 Vector Spaces 1. Not a vector space. Axiom (vi) is not satisfied. 2. Not a vector space. Axiom (i) is not satisfied. 3. Not a vector space. Axiom (x) is not satisfied. 4. A vector space 5. A vector space 6. A vector space 7. Not a vector space. Axiom (ii) is not satisfied. 8. A vector space 9. A vector space 10. Not a vector space. Axiom (i) is not satisfied. 11. A subspace 12. Not a subspace. Axiom (i) is not satisfied. 13. Not a subspace. Axiom (ii) is not satisfied. 14. A subspace 15. A subspace 16. A subspace 17. A subspace 18. A subspace 19. Not a subspace. Neither axioms (i) nor (ii) are satisfied. 20. A subspace 21. Let (x1 , y1 , z1 ) and (x2 , y2 , z2 ) be in S. Then (x1 , y1 , z1 ) + (x2 , y2 , z2 ) = (at1 , bt1 , ct1 ) + (at2 , bt2 , ct2 ) = (a(t1 + t2 ), b(t1 + t2 ), c(t1 + t2 )) is in S. Also, for (x, y, z) in S then k(x, y, z) = (kx, ky, kz) = (a(kt), b(kt), c(kt)) is also in S. 22. Let (x1 , y1 , z1 ) and (x2 , y2 , z2 ) be in S. Then ax1 + by1 + cz1 = 0 and ax2 + by2 + cz2 = 0. Adding gives a(x1 + x2 ) + b(y1 + y2 ) + c(z1 + z2 ) = 0 and so (x1 , y1 , z1 ) + (x2 , y2 , z2 ) = (x1 + x2 , y1 + y2 , z1 + z2 ) is in S. Also, for (x, y, z) then ax + by + cz = 0 implies k(ax + by + cz) = k · 0 = 0 and a(kx) + b(ky) + c(kz) = 0. this means k(x, y, z) = (kx, ky, kz) is in S. 23. (a) c1 u1 + c2 u2 + c3 u3 = 0 if and only if c1 + c2 + c3 = 0, c2 + c3 = 0, c3 = 0. The only solution of this system is c1 = 0, c2 = 0, c3 = 0. (b) Solving the system c1 + c2 + c3 = 3, c2 + c3 = −4, c3 = 8 gives c1 = 7, c2 = −12, c3 = 8. Thus a = 7u1 − 12u2 + 8u3 . 24. (a) The assumption c1 p1 + c2 p2 = 0 is equivalent to (c1 + c2 )x + (c1 − c2 ) = 0. Thus c1 + c2 = 0, c1 − c2 = 0. The only solution of this system is c1 = 0, c2 = 0. (b) Solving the system c1 + c2 = 5, c1 − c2 = 2 gives c1 = 7 2 , c2 = 25. Linearly dependent since ⟨−6, 12⟩ = − 23 ⟨4, −8⟩ 26. Linearly dependent since 2⟨1, 1⟩ + 3⟨0, 1⟩ + (−1)⟨2, 5⟩ = ⟨0, 0⟩ 359 3 2 . Thus p(x) = 27 p1 (x) + 23 p2 (x) 7.6 Vector Spaces 27. Linearly independent 28. Linearly dependent since for all x (1) · 1 + (−2)(x + 1) + (1)(x + 1)2 + (−1)x2 = 0. 29. f is discontinuous at x = −1 and at x = −3. " 2π #2π # x sin x dx = (−x cos x + sin x) # = −2π 30. (x, sin x) = 0 0 * #2π 1 3 ## 2π 3 8 3 2 2 x dx = x # = π and so ∥x∥ = 2 31. ∥x∥ = . Now 3 3 3 0 0 &# % " 2π " 1 2π 1 1 #2π sin2 x dx = ∥ sin x∥2 = (1 − cos 2x) dx = x − sin 2x # = π 2 0 2 2 0 0 √ and so ∥ sin x∥ = π . " 2π 32. A basis could be 1, x, ex cos 3x, ex sin 3x. 33. We need to show that Span{x1 , x2 , . . . , xn } is closed under vector addition and scalar multiplication. Suppose u and v are in Span{x1 , x2 , . . . , xn }. Then u = a1 x1 + a2 x2 + · · · + an xn and v = b1 x1 + b2 x2 + · · · + bn xn , so that u + v = (a1 + b1 )x1 + (a2 + b2 )x2 + · · · + (an + bn )xn , which is in Span{x1 , x2 , . . . , xn }. Also, for any real number k, ku = k(a1 x1 + a2 x2 + · · · + an xn ) = ka1 x1 + ka2 x2 + · · · + kan xn , which is in Span{x1 , x2 , . . . , xn }. Thus, Span{x1 , x2 , . . . , xn } is a subspace of V. 34. R2 is not a subspace of either R3 or R4 and R3 is not a subspace of R4 . The vectors in R2 are ordered pairs, while the vectors in R3 are ordered triples. 35. Since a basis for M22 is B= +( 1 0 ) ( 0 , 0 0 0 ) ( 0 , 1 0 1 ) ( 0 , 0 0 0 0 1 ), , the dimension of M22 is 4. 36. To show that the set of nonzero orthogonal vectors is linearly independent we set c1 v1 + c2 v2 + · · · + cn vn = 0. For 0 ≤ i ≤ n, (c1 v1 + c2 v2 + · · · + ci vi · · · + cn vn ) · vi = c1 v1 · vi + c2 v2 · vi + · · · + ci vi · vi · · · + cn vn · vi = ci ||vi ||2 , so ci ||vi ||2 = 0 because (c1 v1 + c2 v2 + · · · + ci vi · · · + cn vn ) · vi = 0 · vi = 0. Since vi is a nonzero vector, ci = 0. Thus, the assumption that c1 v1 + c2 v2 + · · · + cn vn = 0 leads to c1 = c2 = · · · = cn = 0, and the set is linearly independent. 37. We verify the four properties: (i) (u, v) = u1 v1 + 4u2 v2 = v1 u1 + 4v2 u2 = (v, u) (ii) (ku, v) = (ku1 )v1 + 4(ku2 )v2 = k(u1 v1 + 4u2 v2 ) = k(u, v) (iii) (u, u) = u21 + 4ku22 > 0 for u ̸= 0. Furthermore, u21 + 4ku22 = 0 if and only if u1 = 0 and u2 = 0, or equivalently, u = 0. (iv) (u, v + w) = u1 (v1 + w1 ) + 4u2 (v2 + w2 ) = (u1 v1 + 4u2 v2 ) + (u1 w1 + 4u2 w2 ) = (u, v) + (u, w) 360 7.7 Gram-Schmidt Orthogonalization Process 38. (a) Let u = ⟨2, 1⟩ and v = ⟨2, −1⟩ be nonzero vectors in R2 . With respect to the standard inner or dot product on R2 , u · v = ⟨2, 1⟩ · ⟨2, −1⟩ = 2 · 2 + 1 · (−1) = 3. We see that u and v are not orthogonal with respect to that inner product. But using the inner product in Problem 37, we have (u, v) = 2 · 2 + 4(1) · (−1) = 0, and so u and v are orthogonal with respect to that inner product. (b) Consider f (x) = sin x and g(x) = cos x in C[0, 2π]. Since " 2π " #2π 1 1 1 2π # sin 2x dx = − cos 2x# = − (1 − 1) = 0, sin x cos x dx = 2 4 4 0 0 0 these functions are orthogonal in C[0, 2π]. EXERCISES 7.7 Gram-Schmidt Orthogonalization Process 5 5 12 1. Letting w1 = ⟨ 12 13 , 13 ⟩ and w2 = ⟨ 13 , − 13 ⟩, we have % &% & % &% & 12 5 5 12 w1 · w2 = + − = 0, 13 13 13 13 so the vectors are orthogonal. Also, -% & -% & % &2 % &2 2 2 12 5 5 12 ||w1 || = + = 1 and ||w2 || = + − = 1, 13 13 13 13 so the basis is orthonormal. To express u = ⟨4, 2⟩ in terms of w1 and w2 we compute . / % & % & 12 5 12 5 58 u · w1 = ⟨4, 2⟩ · = (4) + (2) = , 13 13 13 13 13 / % & % & . 12 5 12 4 5 ,− = (4) + (2) − =− , u · w2 = ⟨4, 2⟩ · 13 13 13 13 13 so 58 4 w1 − w2 . 13 13 √ √ √ √ √ √ √ √ 2. Letting w1 = ⟨1/ 3, 1/ 3, −1/ 3⟩, w2 = ⟨0, −1/ 2, −1/ 2⟩, and w3 = ⟨−2/ 6, 1/ 6, −1/ 6⟩, we have & % &% & % &% & % 1 1 1 1 1 (0) + √ −√ + −√ −√ =0 w1 · w2 = √ 3 3 2 3 2 % &% & % &% & % &% & 1 1 1 2 1 1 √ w1 · w3 = √ −√ + √ + −√ −√ =0 3 6 3 6 3 6 % & % &% & % &% & 2 1 1 1 1 √ w2 · w3 = (0) − √ + −√ + −√ −√ = 0, 6 2 6 2 6 u= 361 7.7 Gram-Schmidt Orthogonalization Process so the vectors are orthogonal. Also, ||w1 || = -% 1 √ 3 &2 % &2 &2 % &2 1 1 1 2 + + −√ = 1, ||w2 || = 0 + − √ + −√ = 1, 3 2 2 -% &2 % &2 % &2 1 1 2 + √ + −√ = 1, −√ and ||w3 || = 6 6 6 % 1 √ 3 &2 % so the basis is orthonormal. To express u = ⟨5, −1, 6⟩ in terms of w1 , w2 , and w3 we compute . / % & % & % & 1 1 2 1 1 1 1 √ , √ , −√ = (5) √ + (−1) √ + (6) − √ = −√ 3 3 3 3 3 3 3 . / % & % & 1 1 1 5 1 u · w2 = ⟨5, −1, 6⟩ · 0, − √ , − √ = (5)(0) + (−1) − √ + (6) − √ = −√ 2 2 2 2 2 . / % & % & % & 2 1 2 1 17 1 1 u · w3 = ⟨5, −1, 6⟩ · − √ , √ , − √ = (5) − √ + (−1) √ + (6) − √ = −√ 6 6 6 6 6 6 6 u · w1 = ⟨5, −1, 6⟩ · so 2 5 17 u = − √ w1 − √ w2 − √ w3 . 3 2 6 Since the basis vectors in Problems 3 and 4 are orthogonal but not orthonormal, the result of Theorem 7.5 must be slightly modified to read u= u · w2 u · wn u · w1 w1 + w2 + · · · + wn . 2 2 ||w1 || ||w2 || ||wn ||2 The proof is very similar to that given in the text for Theorem 7.5. 3. Letting w1 = ⟨1, 0, 1⟩, w2 = ⟨0, 1, 0⟩, and w3 = ⟨−1, 0, 1⟩ we have w1 · w2 = (1)(0) + (0)(1) + (1)(0) = 0 w1 · w3 = (1)(−1) + (0)(0) + (1)(1) = 0 w2 · w3 = (0)(−1) + (1)(0) + (0)(1) = 0 so the vectors are orthogonal. We also compute ||w1 ||2 = 12 + 02 + 12 = 2 ||w2 ||2 = 02 + 12 + 02 = 1 ||w3 ||2 = (−1)2 + 02 + 12 = 2 and, with u = ⟨10, 7, −13⟩, u · w1 = (10)(1) + (7)(0) + (−13)(1) = −3 u · w2 = (10)(0) + (7)(1) + (−13)(0) = 7 u · w3 = (10)(−1) + (7)(0) + (−13)(1) = −23. Then, using the result given before the solution to this problem, we have 3 23 u = − w1 + 7w2 − w3 . 2 2 362 7.7 Gram-Schmidt Orthogonalization Process 4. Letting w1 = ⟨2, 1, −2, 0⟩, w2 = ⟨1, 2, 2, 1⟩, w3 = ⟨3, −4, 1, 3⟩, and w4 = ⟨5, −2, 4, −9⟩ we have w1 · w2 = (2)(1) + (1)(2) + (−2)(2) + (0)(1) = 0 w1 · w3 = (2)(3) + (1)(−4) + (−2)(1) + (0)(3) = 0 w1 · w4 = (2)(5) + (1)(−2) + (−2)(4) + (0)(−9) = 0 w2 · w3 = (1)(3) + (2)(−4) + (2)(1) + (1)(3) = 0 w2 · w4 = (1)(5) + (2)(−2) + (2)(4) + (1)(−9) = 0 w3 · w4 = (3)(5) + (−4)(−2) + (1)(4) + (3)(−9) = 0 so the vectors are orthogonal. We also compute ||w1 ||2 = 22 + 12 + (−2)2 + 02 = 9 ||w2 ||2 = 12 + 22 + 22 + 12 = 10 ||w3 ||2 = 32 + (−4)2 + 12 + 32 = 35 ||w4 ||2 = 52 + (−2)2 + 42 + (−9)2 = 126 and, with u = ⟨1, 2, 4, 3⟩, u · w1 = (1)(2) + (2)(1) + (4)(−2) + (3)(0) = −4 u · w2 = (1)(1) + (2)(2) + (4)(2) + (3)(1) = 16 u · w3 = (1)(3) + (2)(−4) + (4)(1) + (3)(3) = 8 u · w4 = (1)(5) + (2)(−2) + (4)(4) + (3)(−9) = −10. Then, using the result given before the solution to this problem, we have 4 8 8 5 u = − w1 + w2 + w3 − w4 . 9 5 35 63 5. (a) We have u1 = ⟨−3, 2⟩ and u2 = ⟨−1, −1⟩. Taking v1 = u1 = ⟨−3, 2⟩, and using u2 · v1 = 1 and v1 · v1 = 13 we obtain / . u2 · v1 1 10 15 . v2 = u2 − v1 = ⟨−1, −1⟩ − ⟨−3, 2⟩ = − , − v1 · v1 13 13 13 15 ′ ′ Thus, an orthogonal basis is {⟨−3, 2⟩, ⟨− 10 13 , − 13 ⟩} and an orthonormal basis is {w1 , w2 }, where / . 3 2 1 1 ′ w1 = ⟨−3, 2⟩ = √ ⟨−3, 2⟩ = − √ , √ ||⟨−3, 2⟩|| 13 13 13 and w2′ 1 = 10 ||⟨− 13 , − 15 13 ⟩|| . 10 15 − ,− 13 13 / . / / . 10 15 3 2 − ,− . = = −√ , −√ 13 13 5/ 13 13 13 1 √ (b) We have u1 = ⟨−3, 2⟩ and u2 = ⟨−1, −1⟩. Taking v1 = u2 = ⟨−1, −1⟩, and using u1 · v1 = 1 and v1 · v1 = 2 we obtain v2 = u1 − u1 · v1 1 v1 = ⟨−3, 2⟩ − ⟨−1, −1⟩ = v1 · v1 2 . 5 5 − , 2 2 / . Thus, an orthogonal basis is {⟨−1, −1⟩, ⟨− 25 , 52 ⟩} and an orthonormal basis is {w3′′ , w4′′ }, where / . 1 1 1 1 ′′ ⟨−1, −1⟩ = √ ⟨−1, −1⟩ = − √ , − √ w3 = ||⟨−1, −1⟩|| 2 2 2 and w4′′ 1 = 5 5 ||⟨− 2 , 2 ⟩|| / . / . / . 1 5 5 1 1 5 5 = √ = −√ , √ − , . − , 2 2 2 2 5/ 2 2 2 363 7.7 Gram-Schmidt Orthogonalization Process (c) 1 4 u -4 w1 2 -2 2 4 -1 1 w4 0.5 -0.5 v-2 0.5 1 -1 0.5 -0.5 -0.5 w2 -4 0.5 1 -0.5 w3 -1 -1 6. (a) We have u1 = ⟨−3, 4⟩ and u2 = ⟨−1, 0⟩. Taking v1 = u1 = ⟨−3, 4⟩, and using u2 · v1 = 3 and v1 · v1 = 25 we obtain . / 3 16 12 u2 · v1 v1 = ⟨−1, 0⟩ − ⟨−3, 4⟩ = − , − v2 = u2 − . v1 · v1 25 25 25 12 ′ ′ Thus, an orthogonal basis is {⟨−3, 4⟩, ⟨− 16 25 , − 25 ⟩} and an orthonormal basis is {w1 , w2 }, where . / 1 3 4 1 ′ ⟨−3, 4⟩ = ⟨−3, 4⟩ = − , w1 = ||⟨−3, 4⟩|| 5 5 5 and w2′ 1 = 16 ||⟨− 25 , − 12 25 ⟩|| . 16 12 − ,− 25 25 / 1 = 4/5 . / . / 16 12 4 3 − ,− = − ,− . 25 25 5 5 (b) We have u1 = ⟨−3, 4⟩ and u2 = ⟨−1, 0⟩. Taking v1 = u2 = ⟨−1, 0⟩, and using u1 · v1 = 3 and v1 · v1 = 1 we obtain v2 = u1 − 3 u1 · v1 v1 = ⟨−3, 4⟩ − ⟨−1, 0⟩ = ⟨0, 4⟩ . v1 · v1 1 Thus, an orthogonal basis is {⟨−1, 0⟩, ⟨0, 4⟩} and an orthonormal basis is {w3′′ , w4′′ }, where w3′′ = 1 1 ⟨−1, 0⟩ = ⟨−1, 0⟩ = ⟨−1, 0⟩ ||⟨−1, 0⟩|| 1 and w4′′ = (c) u 1 1 ⟨0, 4⟩ = ⟨0, 4⟩ = ⟨0, 1⟩ . ||⟨0, 4⟩|| 4 1 4 1 w1 0.5 2 0.5 w3 v -4 w4 -2 2 4 -1 -2 -0.5 w2 -4 1 0.5 -1 -0.5 0.5 -0.5 -0.5 -1 -1 1 7. (a) We have u1 = ⟨1, 1⟩ and u2 = ⟨1, 0⟩. Taking v1 = u1 = ⟨1, 1⟩, and using u2 · v1 = 1 and v1 · v1 = 2 we obtain / . u2 · v1 1 1 1 v2 = u2 − . v1 = ⟨1, 0⟩ − ⟨1, 1⟩ = ,− v1 · v1 2 2 2 Thus, an orthogonal basis is {⟨1, 1⟩, ⟨ 21 , − 12 ⟩} and an orthonormal basis is {w1′ , w2′ }, where / . 1 1 1 1 ⟨1, 1⟩ = √ ⟨1, 1⟩ = √ , √ w1′ = ||⟨1, 1⟩|| 2 2 2 and w2′ = 1 ||⟨ √12 , − √12 ⟩|| . 1 1 √ , −√ 2 2 / = 1 1 364 . 1 1 √ , −√ 2 2 / = . 1 1 √ , −√ 2 2 / . 7.7 Gram-Schmidt Orthogonalization Process (b) We have u1 = ⟨1, 1⟩ and u2 = ⟨1, 0⟩. Taking v1 = u2 = ⟨1, 0⟩, and using u1 · v1 = 1 and v1 · v1 = 1 we obtain v2 = u1 − u1 · v1 1 v1 = ⟨1, 1⟩ − ⟨1, 0⟩ = ⟨0, 1⟩ . v1 · v1 1 Thus, an orthogonal basis is {⟨1, 0⟩, ⟨0, 1⟩}, which is also an orthonormal basis. (c) 2 1 1.5 1 u 0.5 -2 1 w1 0.5 w4 0.5 w3 v -1 1 -1 2 -0.5 0.5 -0.5 -1 -0.5 w2 -0.5 -1 1 1 0.5 -0.5 -1.5 -1 -2 -1 8. (a) We have u1 = ⟨5, 7⟩ and u2 = ⟨1, −2⟩. Taking v1 = u1 = ⟨5, 7⟩, and using u2 · v1 = −9 and v1 · v1 = 74 we obtain v2 = u2 − 9 u2 · v1 v1 = ⟨1, −2⟩ − ⟨5, 7⟩ = v1 · v1 74 . 119 85 ,− 74 74 / . 85 ′ ′ Thus, an orthogonal basis is {⟨5, 7⟩, ⟨ 119 74 , − 74 ⟩} and an orthonormal basis is {w1 , w2 }, where / . 1 1 5 7 w1′ = ⟨5, 7⟩ = √ ⟨5, 7⟩ = √ , √ ||⟨5, 7⟩|| 74 74 74 and w2′ = 1 85 , ||⟨ 119 74 − 74 ⟩|| . 119 85 ,− 74 74 / = 1 √ 17/ 74 . 119 85 ,− 74 74 / = . 7 5 √ , −√ 74 74 / . (b) We have u1 = ⟨5, 7⟩ and u2 = ⟨1, −2⟩. Taking v1 = u2 = ⟨1, −2⟩, and using u1 · v1 = −9 and v1 · v1 = 5 we obtain . / 34 17 9 u1 · v1 v1 = ⟨5, 7⟩ − ⟨1, −2⟩ = v2 = u1 − , . v1 · v1 5 5 5 17 ′′ ′′ Thus, an orthogonal basis is {⟨1, −2⟩, ⟨ 34 5 , 5 ⟩} and an orthonormal basis is {w3 , w4 }, where / . 1 1 1 2 ′′ ⟨1, −2⟩ = √ ⟨1, −2⟩ = √ , − √ w3 = ||⟨1, −2⟩|| 5 5 5 and w4′′ (c) 1 = 34 17 ||⟨ 5 , 5 ⟩|| . 34 17 , 5 5 8 1 √ = 17/ 5 . 34 17 , 5 5 / = . 2 1 √ ,√ 5 5 1 u 6 / 4 0.5 / . 1 w1 0.5 w4 2 -7.5 -5 -2.5 -2 -4 2.5 5 7.5 v -1 -0.5 0.5 -0.5 -6 -8 w2 -1 1 -1 -0.5 0.5 1 -0.5 -1 w3 9. We have u1 = ⟨1, 1, 0⟩, u2 = ⟨1, 2, 2⟩, and u3 = ⟨2, 2, 1⟩. Taking v1 = u1 = ⟨1, 1, 0⟩ and using u2 · v1 = 3 and v1 · v1 = 2 we obtain . / 3 1 1 u2 · v1 v1 = ⟨1, 2, 2⟩ − ⟨1, 1, 0⟩ = − , , 2 . v2 = u2 − v1 · v1 2 2 2 365 7.7 Gram-Schmidt Orthogonalization Process Next, using u3 · v1 = 4, u3 · v2 = 2, and v2 · v2 = 92 , we obtain u3 · v2 4 u3 · v1 2 v1 − v2 = ⟨2, 2, 1⟩ − ⟨1, 1, 0⟩ − v3 = u3 − v1 · v1 v2 · v2 2 9/2 . / . / 1 1 2 1 2 − , ,2 = ,− , . 2 2 9 9 9 Thus, an orthogonal basis is B′ = + . / . /, 1 1 2 2 1 ⟨1, 1, 0⟩ , − , , 2 , ,− , , 2 2 9 9 9 and an orthonormal basis is ′′ B = +. /, / . / . 1 2 1 2 1 1 1 4 √ , √ ,0 , − √ , √ , √ ,− , . , 3 3 3 2 2 3 2 3 2 3 2 10. We have u1 = ⟨−3, 1, 1⟩, u2 = ⟨1, 1, 0⟩, and u3 = ⟨−1, 4, 1⟩. Taking v1 = u1 = ⟨−3, 1, 1⟩ and using u2 · v1 = −2 and v1 · v1 = 11 we obtain . / 5 13 2 −2 u2 · v1 v1 = ⟨1, 1, 0⟩ − ⟨−3, 1, 1⟩ = , , v2 = u2 − . v1 · v1 11 11 11 11 Next, using u3 · v1 = 8, u3 · v2 = v3 = u3 − 49 11 , and v2 · v2 = 18 11 , we obtain u3 · v1 u3 · v2 8 49/11 v1 − v2 = ⟨−1, 4, 1⟩ − ⟨−3, 1, 1⟩ − v1 · v1 v2 · v2 11 18/11 . 5 13 2 , , 11 11 11 / = . / 1 1 2 − , ,− . 18 18 9 Thus, an orthogonal basis is / . /, + . 2 1 1 5 13 2 , , ,− , − , , B = ⟨−3, 1, 1⟩ , 11 11 11 18 18 9 ′ and an orthonormal basis is +. / . / . /, 3 1 1 13 2 1 4 5 1 ′′ √ , √ , √ B = −√ , √ , √ , , − √ , √ , √ . 11 11 11 3 22 3 22 3 22 3 2 3 2 3 2 11. We have u1 = ⟨ 21 , 12 , 1⟩, u2 = ⟨−1, 1, − 12 ⟩, and u3 = ⟨−1, 12 , 1⟩. Taking v1 = u1 = ⟨ 21 , 21 , 1⟩ and using u2 ·v1 = − 12 and v1 · v1 = 23 we obtain . / . / . / 1 5 7 1 −1/2 1 1 u2 · v1 v1 = −1, 1, − , ,1 = − , ,− − . v2 = u2 − v1 · v1 2 3/2 2 2 6 6 6 Next, using u3 · v1 = 34 , u3 · v2 = 45 , and v2 · v2 = 25 12 , we obtain . . . / . / / / u3 · v2 1 5 7 1 3 3/4 1 1 5/4 9 3 u3 · v1 v1 − v2 = −1, , 1 − − , ,− = − ,− , . , ,1 − v3 = u3 − v1 · v1 v2 · v2 2 3/2 2 2 25/12 6 6 6 4 20 5 Thus, an orthogonal basis is ′ B = +. / . / . /, 1 1 5 7 1 9 3 3 , ,1 , − , ,− , − ,− , , 2 2 6 6 6 4 20 5 and an orthonormal basis is +. / . / . /, 1 1 2 7 1 3 4 1 1 ′′ √ ,√ ,√ B = , −√ , √ , − √ , −√ , − √ , √ . 6 6 6 3 5 3 5 3 2 5 2 5 2 12. We have u1 = ⟨1, 1, 1⟩, u2 = ⟨9, −1, 1⟩, and u3 = ⟨−1, 4, −2⟩. Taking v1 = u1 = ⟨1, 1, 1⟩ and using u2 · v1 = 9 and v1 · v1 = 3 we obtain v2 = u2 − 9 u2 · v1 v1 = ⟨9, −1, 1⟩ − ⟨1, 1, 1⟩ = ⟨6, −4, −2⟩ . v1 · v1 3 366 7.7 Gram-Schmidt Orthogonalization Process Next, using u3 · v1 = 1, u3 · v2 = −18, and v2 · v2 = 56, we obtain u3 · v2 1 −18 u3 · v1 v1 − v2 = ⟨−1, 4, −2⟩ − ⟨1, 1, 1⟩ − ⟨6, −4, −2⟩ = v3 = u3 − v1 · v1 v2 · v2 3 56 . 25 50 125 , ,− 42 21 42 / . Thus, an orthogonal basis is ′ B = + ⟨1, 1, 1⟩ , ⟨6, −4, −2⟩ , . 25 50 125 , ,− 42 21 42 /, , and an orthonormal basis is / . / . /, +. 3 1 1 1 2 1 4 5 1 √ ,√ ,√ B ′′ = , √ , −√ , −√ , √ , √ , −√ . 3 3 3 14 14 14 42 42 42 13. We have u1 = ⟨1, 5, 2⟩, and u2 = ⟨−2, 1, 1⟩. Taking v1 = u1 = ⟨1, 5, 2⟩ and using u2 · v1 = 5 and v1 · v1 = 30 we obtain / . u2 · v1 5 13 1 2 v2 = u2 − . v1 = ⟨−2, 1, 1⟩ − ⟨1, 5, 2⟩ = − , , v1 · v1 30 6 6 3 0 1 23 1 2 Thus, an orthogonal basis is B ′ = ⟨1, 5, 2⟩ , − 13 , and an orthonormal basis is 6 , 6 , 3 / . /, +. 13 5 2 1 4 1 ′′ √ ,√ ,√ , −√ . ,√ ,√ B = 30 30 30 186 186 186 14. We have u1 = ⟨1, 2, 3⟩, and u2 = ⟨3, 4, 1⟩. Taking v1 = u1 = ⟨1, 2, 3⟩ and using u2 · v1 = 14 and v1 · v1 = 14 we obtain v2 = u2 − u2 · v1 14 v1 = ⟨3, 4, 1⟩ − ⟨1, 2, 3⟩ = ⟨2, 2, −2⟩ . v1 · v1 14 Thus, an orthogonal basis is B ′ = {⟨1, 2, 3⟩ , ⟨2, 2, −2⟩} , and an orthonormal basis is / . /, +. 1 2 3 1 1 1 ′′ √ ,√ ,√ , √ , √ , −√ . B = 14 14 14 3 3 3 15. We have u1 = ⟨1, −1, 1, −1⟩, and u2 = ⟨1, 3, 0, 1⟩. Taking v1 = u1 = ⟨1, −1, 1, −1⟩ and using u2 · v1 = −3 and v1 · v1 = 4 we obtain . / 7 9 3 1 −3 u2 · v1 v1 = ⟨1, 3, 0, 1⟩ − ⟨1, −1, 1, −1⟩ = , , , . v2 = u2 − v1 · v1 4 4 4 4 4 0 1 23 Thus, an orthogonal basis is B ′ = ⟨1, −1, 1, −1⟩ , 47 , 94 , 34 , 41 , and an orthonormal basis is +. / . /, 1 9 3 1 1 1 1 7 ′′ √ √ √ √ B = ,− , ,− , . , , , 2 2 2 2 2 35 2 35 2 35 2 35 16. We have u1 = ⟨4, 0, 2, −1⟩, u2 = ⟨2, 1, −1, 1⟩, and u3 = ⟨1, 1, −1, 0⟩. Taking v1 = u1 = ⟨4, 0, 2, −1⟩ and using u2 · v1 = 5 and v1 · v1 = 21 we obtain 5 u2 · v1 v1 = ⟨2, 1, −1, 1⟩ − ⟨4, 0, 2, −1⟩ = v2 = u2 − v1 · v1 21 Next, using u3 · v1 = 2, u3 · v2 = 74 21 , and v2 · v2 = 122 21 . 22 31 26 , 1, − , 21 21 21 / . , we obtain u3 · v2 u3 · v1 v1 − v2 v1 · v1 v2 · v2 . / . / 2 22 74/21 31 26 1 24 18 40 = ⟨1, 1, −1, 0⟩ − ⟨4, 0, 2, −1⟩ − , 1, − , = − , ,− ,− . 21 122/21 21 21 21 61 61 61 61 v3 = u3 − Thus, an orthogonal basis is / . /, + . 31 26 1 24 18 40 22 , 1, − , , − , ,− ,− , B ′ = ⟨4, 0, 2, −1⟩ , 21 21 21 61 61 61 61 367 7.7 Gram-Schmidt Orthogonalization Process and an orthonormal basis is 4. / . / 4 22 2 1 21 31 26 ′′ √ , 0, √ , − √ B = , √ , ,√ , −√ ,√ 21 21 21 2562 2562 2562 2562 . /5 1 24 18 40 −√ ,√ , −√ , −√ . 2501 2501 2501 2501 17. We have u1 = 1, u2 = x, and u3 = x2 . Taking v1 = u1 = 1 and using " " 1 1 · x2 dx = 0 and (v1 , v1 ) = (u2 , v1 ) = 1 x · x dx = 2 −1 −1 we obtain v2 = u2 − (u2 , v1 ) 0 v1 = x − x = x. (v1 , v1 ) 2 Next, using 1 2 x · 1 dx = , (u3 , v1 ) = 3 −1 " 2 " (u3 , v2 ) = 1 2 x · x dx = 0, and (v2 , v2 ) = " 1 x · x dx = −1 −1 2 , 3 we obtain (u3 , v2 ) 2/3 0 1 (u3 , v1 ) v1 − v2 = x2 − 1− x = x2 − . (v1 , v1 ) (v2 , v2 ) 2 2/3 3 3 0 Thus, an orthogonal basis is B ′ = 1, x, x2 − 13 . v3 = u3 − 18. We have u1 = x2 − x, u2 = x2 + 1, and u3 = 1 − x2 . Taking v1 = u1 = x2 − x and using " 1 " 1 16 16 2 2 (x + 1)(x − x)dx = (u2 , v1 ) = (x2 − x)(x2 − x)dx = and (v1 , v1 ) = 15 15 −1 −1 we obtain v2 = u2 − (u2 , v1 ) 16/15 2 v1 = x2 + 1 − (x − x) = x + 1. (v1 , v1 ) 16/15 Next, using (u3 , v1 ) = " 1 −1 (1 − x2 )(x2 − x)dx = and (v2 , v2 ) = 4 , 15 (u3 , v2 ) = " −1 1 " 1 (x + 1)(x + 1)dx = −1 (1 − x2 )(x + 1)dx = 4 , 3 8 , 3 we obtain (u3 , v2 ) 4/15 2 4/3 5 1 1 (u3 , v1 ) v1 − v2 = 1 − x2 − (x − x) − (x + 1) = − x3 − x + . (v1 , v1 ) (v2 , v2 ) 16/15 8/3 4 4 2 0 3 Thus, an orthogonal basis is B ′ = x2 − x, x + 1, − 54 x3 − 41 x + 21 . v3 = u3 − 19. Using the solution of Problem 17 and computing " 1 ||v1 ||2 = (v1 , v1 ) = 1 · 1 dx = 2, ||v2 ||2 = (v2 , v2 ) = −1 and ||v3 ||2 = (v3 , v3 ) = " 1 −1 % x2 − 368 " 1 x · x dx = −1 1 3 &% x2 − 1 3 & dx = 8 , 45 2 , 3 7.7 Gram-Schmidt Orthogonalization Process we see that an orthonormal basis is 5 + 4 % &, 1 1 x 3 15 x2 − 1/3 1 ′′ = √ , √ x, √ . x2 − , $ B = √ ,$ 3 2 2 6 2 10 2/3 8/45 20. Using the solution of Problem 18 and computing " 1 16 2 ||v1 || = (v1 , v1 ) = (x2 − x)(x2 − x)dx = , 15 −1 and 2 ||v3 || = (v3 , v3 ) = we see that an orthonormal basis is 4√ B ′′ = " 1 −1 % 2 ||v2 || = (v2 , v2 ) = 5 1 1 − x3 − x + 4 4 2 " 1 (x + 1)(x + 1)dx = −1 &% 5 1 1 − x3 − x + 4 4 2 & dx = 8 , 3 1 , 3 5 √ 3 15 2 3 (x − x), √ (x + 1), (−5x2 − x + 2) . 4 4 2 6 √ √ √ 21. Using w1 = 1/ 2, w2 = 3x/ 6, and w3 = (15/2 10)(x2 − 1/3), and computing " 1 √ 1 (p, w1 ) = (9x2 − 6x + 5) √ dx = 8 2, 2 −1 " 1 √ 3 (9x2 − 6x + 5) √ x dx = −2 6 (p, w2 ) = 6 −1 &) % ( " 1 15 1 12 2 2 dx = √ , x − (9x − 6x + 5) √ (p, w3 ) = 3 2 10 10 −1 we find from Theorem 7.5 √ √ 12 p(x) = 9x2 − 6x + 5 = (p, w1 )w1 + (p, w2 )w2 + (p, w3 )w3 = 8 2 w1 − 2 6 w2 + √ w3 . 10 √ √ √ 22. Using w1 = ( 15/4)(x2 − x), w2 = (3/2 6)(x + 1), and w3 = −( 3/4)(5x2 + x − 2), and computing 9 6√ " 1 7 8 41 15 x2 − x dx = √ , (9x2 − 6x + 5) (p, w1 ) = 4 15 −1 ( ) " 1 √ 3 (p, w2 ) = (9x2 − 6x + 5) √ (x + 1) dx = 3 6 2 6 −1 9 6 √ " 1 1 3 2 2 (5x + x − 2) dx = √ , (9x − 6x + 5) − (p, w3 ) = 4 3 −1 we find from Theorem 7.5 √ 41 1 p(x) = 9x2 − 6x + 5 = (p, w1 )w1 + (p, w2 )w2 + (p, w3 )w3 = √ w1 + 3 6 w2 + √ w3 . 15 3 23. Since u3 depends on u1 and u2 we would expect the Gram-Schmidt process to yield a pair of orthogonal vectors v1 and v2 , with a third vector v3 that is 0. This is because u3 lies in the subspace W2 of R3 spanned by u1 and u2 , and hence the projection of u3 onto W2 is u3 itself. In other words, u3 · v2 u3 · v1 u3 · v2 u3 · v1 v1 + v2 so v3 = u3 − v1 + v2 = 0. u3 = projW3 u3 = v1 · v1 v2 · v2 v1 · v1 v2 · v2 To carry out the orthogonalization process we take v1 = u1 = ⟨1, 1, 3⟩. Then, using u2 · v1 = 8 and v1 · v1 = 11 we obtain v2 = u2 − u2 · v1 8 v1 = ⟨1, 4, 1⟩ − ⟨1, 1, 3⟩ = v1 · v1 11 369 . 3 36 13 , ,− 11 11 11 / . 7.7 Gram-Schmidt Orthogonalization Process Next, using u3 · v1 = 2, u3 · v2 = 402 11 , and v2 · v2 = 134 11 , we obtain u3 · v1 u3 · v2 2 402/11 v3 = u3 − v1 − v2 = ⟨1, 10, −3⟩ − ⟨1, 1, 3⟩ − v1 · v1 v2 · v2 11 134/11 . 3 36 13 , ,− 11 11 11 / = ⟨0, 0, 0⟩ . 3 36 3 In this case {v1 , v2 } = {⟨1, 1, 3}, ⟨ 11 , 11 , − 13 11 ⟩} is an orthogonal subset of R containing the third vector u3 = ⟨1, 10, −3⟩. CHAPTER 7 REVIEW EXERCISES 1. True 2. False; the points must be non-collinear. 3. False; since a normal to the plane is ⟨2, 3, −4⟩ which is not a multiple of the direction vector ⟨5, −2, 1⟩ of the line. 4. True 5. True 6. True 7. True 8. True 9. True 10. True; since a × b and c × d are both normal to the plane and hence parallel (unless a × b = 0 or c × d = 0.) 11. 9i + 2j + 2k 12. orthogonal 13. −5(k × j) = −5(−i) = 5i $ 15. (−12)2 + 42 + 62 = 14 14. i · (i × j) = i × k = 0 16. (−1 − 20)i − (−2 − 0)j + (8 − 0)k = −21i + 2j + 8k 17. −6i + j − 7k 18. The coordinates of (1, −2, −10) satisfy the given equation. 19. Writing the line in parametric form, we have x = 1 + t, y = −2 + 3t, z = −1 + 2t. Substituting into the equation of the plane yields (1 + t) + 2(−2 + 3t) − (−1 + 2t) = 13 or t = 3. Thus, the point of intersection is x = 1 + 3 = 4, y = −2 + 3(3) = 7, z = −1 + 2(3) = 5, or (4, 7, 5). 20. |a| = $ √ 1 4 3 1 42 + 32 + (−5)2 = 5 2 ; u = − √ (4i + 3j − 5k) = − √ i − √ j + √ k 5 2 5 2 5 2 2 21. x2 − 2 = 3, x2 = 5; y2 − 1 = 5, y2 = 6; z2 − 7 = −4, z2 = 3; P2 = (5, 6, 3) 22. (5, 1/2, 5/2) √ 23. (7.2)(10) cos 135◦ = −36 2 24. 2b = ⟨−2, 4, 2⟩; 4c = ⟨0, −8, 8⟩; a · (2b + 4c) = ⟨3, 1, 0⟩ · ⟨−2, −4, 10⟩ = −10 25. 12, −8, 6 a·b 1 1 26. cos θ = = √ √ = ; θ = 60◦ |a||b| 2 2 2 √ 1 3 10 27. A = |5i − 4j − 7k| = 2 2 370 CHAPTER 7 REVIEW EXERCISES 28. From 3(x − 3) + 0(y − 6) + (1)(z − (−2)) = 0 we obtain 3x + z = 7. 29. | − 5 − (−3)| = 2 30. parallel: −2c = 5, # #i j # # 31. a × b = # 1 1 # # 1 −2 and b is c = −5/2; orthogonal: 1(−2) + 3(−6) + c(5) = 0, c = 4 # k ## # # # # # # #1 1 # #1 0# # ## 1 0 ## # k = i − j − 3k A unit vector perpendicular to both a # # # j+# i−# 0# = # # 1 −2 # 1 1# −2 1 # 1# a×b 1 1 1 3 (i − j − 3k) = √ i − √ j − √ k. =√ ∥a × b∥ 1+1+9 11 11 11 $ 3 1/2 2 1/2 2 1/4 + 1/4 + 1/6 = ; cos α = = , α ≈ 48.19◦ ; cos β = = , β ≈ 48.19◦ ; 4 3/4 3 3/4 3 1 −1/4 cos γ = = − , γ ≈ 109.47◦ 3/4 3 32. ∥a∥ = 33. compb a = a · b/∥b∥ = ⟨1, 2, −2⟩ · ⟨4, 3, 0⟩/5 = 2 34. compa b = b · a/∥a∥ = ⟨4, 3, 0⟩ · ⟨1, 2, −2⟩/3 = 10/3 proja b = (compa b)a/∥a∥ = (10/3)⟨1, 2, −2⟩/3 = ⟨10/9, 20/9, −20/9⟩ 35. a + b = ⟨1, 2, −2⟩ + ⟨4, 3, 0⟩ = ⟨5, 5, −2⟩ √ compa (a + b) = (a + b) · a/ 1 + 4 + 4 = 13 (a · a + b · a) = 31 [(1 + 4 + 4) + (4 + 6 + 0)] = 1 2 2 19 38 38 proja (a + b) = [compa (a + b)](a/∥a∥) = 19 3 ⟨3, 3, −3⟩ = ⟨ 9 , 9 , − 9 ⟩ 19 3 36. a − b = ⟨1, 2, −2⟩ − ⟨4, 3, 0⟩ = ⟨−3, −1, −2⟩ √ compb (a − b) = (a − b) · b/ 16 + 9 = 51 (a · b − b · b) = 15 [(4 + 6 + 0) − (16 + 9)] = −3 9 projb (a − b) = [compb (a − b)](b/∥b∥) = −3⟨ 45 , 53 , 0⟩ = ⟨− 12 5 , − 5 , 0⟩ 37. Let a = ⟨a, b, c⟩ and r = ⟨x, y, z⟩. Then (a) (r − a) · r = ⟨x − a, y − b, z − c⟩ · ⟨x, y, z⟩ = x2 − ax + y 2 − by + z 2 − zc = 0 implies b c a2 + b2 + c2 a . The surface is a sphere. (x − )2 + (y − )2 + (z − )2 = 2 2 2 4 (b) (r − a) · a = ⟨x − a, y − b, z − c⟩ · ⟨a, b, c⟩ = a(x − a) + b(y − b) + c(z − c) = 0 The surface is a plane. 38. ⟨4, 2, −2⟩ − ⟨2, 4, −3⟩ = ⟨2, −2, 1⟩; ⟨2, 4, −3⟩ − ⟨6, 7, −5⟩ = ⟨−4, −3, 2⟩; ⟨2, −2, 1⟩ · ⟨−4, −3, 2⟩ = 0 The points are the vertices of a right triangle. 39. A direction vector of the given line is ⟨4, −2, 6⟩. A parallel line containing (7, 3, −5) is (x−7)/4 = (y −3)/(−2) = (z + 5)/6. 40. A normal to the plane is ⟨8, 3, −4⟩. The line with this direction vector and through (5, −9, 3) is x = 5 + 8t, y = −9 + 3t, z = 3 − 4t. 41. The direction vectors are ⟨−2, 3, 1⟩ and ⟨2, 1, 1⟩. Since ⟨−2, 3, 1⟩ · ⟨2, 1, 1⟩ = 0, the lines are orthogonal. Solving 1 − 2t = x = 1 + 2s, 3t = y = −4 + s, we obtain t = −1 and s = 1. The point (3, −3, 0) obtained by letting t = −1 and s = 1 is common to the two lines, so they do intersect. 42. Vectors in the plane are ⟨2, 3, 1⟩ and ⟨1, 0, 2⟩. A normal vector is ⟨2, 3, 1⟩ × ⟨1, 0, 2⟩ = ⟨6, −3, −3⟩ = 3⟨2, −1, −1⟩. An equation of the plane is 2x − y − z = 0 43. The lines are parallel with direction vector ⟨1, 4, −2⟩. Since (0, 0, 0) is on the first line and (1, 1, 3) is on the second line, the vector ⟨1, 1, 3⟩ is in the plane. A normal vector to the plane is thus ⟨1, 4, −2⟩ × ⟨1, 1, 3⟩ = ⟨14, −5, −3⟩. An equation of the plane is 14x − 5y − 3z = 0. 371 CHAPTER 7 REVIEW EXERCISES 44. Letting z = t in the equations of the plane and solving −x + y = 4 + 8t, 3x − y = −2t, we obtain x = 2 + 3t, y = 6 + 11t, z = t. Thus, a normal to the plane is ⟨3, 11, 1⟩ and an equation of the plane is 3(x − 1) + 11(y − 7) + (z + 1) = 0 or 3x + 11y + z = 79. √ √ a 10 = √ (i + j) = 5 2 i + 5 2 j; d = ⟨7, 4, 0⟩ − ⟨4, 1, 0⟩ = 3i + 3j ∥a∥ 2 √ √ √ W = F · d = 15 2 + 15 2 = 30 2 N-m √ √ √ √ 46. F = 5 2 i + 5 2 j + 50i = (5 2 + 50)i + 5 2 j; d = 3i + 3j √ √ √ W = 15 2 + 150 + 15 2 = 30 2 + 150 N-m ≈ 192.4 N-m √ √ √ √ √ 47. Since F2 = 200(i + j)/ 2 = 100 2 i + 100 2 j, F3 = F2 − F1 = (100 2 − 200)i + 100 2 j and ! $ √ √ √ ∥F3 ∥ = (100 2 − 200)2 + (100 2)2 = 200 2 − 2 ≈ 153 lb. 45. F = 10 48. Let ∥F1 ∥ = F1 and ∥F2 ∥ = F2 . Then F1 = F1 [(cos 45◦ )i + (sin 45◦ )j] and F2 = F2 [(cos 120◦ )i + (sin 120◦ )j], or √ F1 = F1 ( √12 i + √12 j) and F2 = F2 (− 21 i + 23 j). Since w + F1 + F2 = 0, √ √ 1 1 1 1 1 1 3 3 F1 ( √ i + √ j) + F2 (− i + j) = 50j, ( √ F1 − F2 )i + ( √ F1 + F2 )j = 50j 2 2 2 2 2 2 2 2 and √ 1 1 1 3 √ F1 − F2 = 0, √ F1 + F2 = 50. 2 2 2 2 √ √ √ Solving, we obtain F1 = 25( 6 − 2 ) ≈ 25.9 lb and F2 = 50( 3 − 1) ≈ 36.6 lb. 49. Not a vector space. Axiom (viii) is not satisfied. 50. The vectors are linearly independent. The only solution of the system c1 = 0, c1 + 2c2 + c3 = 0, 2c1 + 3c2 − c3 = 0 is c1 = 0, c2 = 0, c3 = 0. 51. Let p1 and p2 be in Pn such that d 2 p2 d 2 p1 = 0 and = 0. Since dx2 dx2 d 2 p1 d 2 p2 d2 d 2 p1 d2 + = 2 (p1 + p2 ) and 0 = k = 2 (kp1 ) 2 2 2 dx dx dx dx dx we conclude that the set of polynomials with the given property is a subspace of Pn . A basis for the subspace is 1, x. 0= 52. The intersection W1 ∩ W2 is a subspace of V . If x and y are in W1 ∩ W2 then x and y are in each subspace and so x + y is in each subspace. That is, x + y is in W1 ∩ W2 . Similarly, if x is in W1 ∩ W2 then x is in each subspace and so kx is in each subspace. That is, kx is in W1 ∩ W2 for any scalar k. # # The union W1 ∪ W2 is generally not a subspace. For example, W1 = {⟨x, y⟩ # y = x} and W2 = {⟨x, y⟩ # y = 2x} are subspaces of R2 . Now ⟨1, 1⟩ is in W1 and ⟨1, 2⟩ is in W2 but ⟨1, 1⟩ + ⟨1, 2⟩ = ⟨2, 3⟩ is not in W1 ∪ W2 . 372 8 Matrices EXERCISES 8.1 Matrix Algebra 1. 2 × 4 2. 3 × 2 3. 3 × 3 4. 1 × 3 6. 8 × 1 7. Not equal 8. Not equal 9. Not equal 11. Solving x = y − 2, y = 3x − 2 we obtain x = 2, y = 4. 12. Solving x2 = 9, y = 4x we obtain x = 3, y = 12 and x = −3, t = −12. 13. c23 = 2(0) − 3(−3) = 9; c12 = 2(3) − 3(−2) = 12 14. c23 = 2(1) − 3(0) = 2; c12 = 2(−1) − 3(0) = −2 " ! ! " 2 11 4−2 5+6 = 15. (a) A + B = −6 + 8 9 − 10 2 −1 " " ! ! −6 1 −2 − 4 6−5 = (b) B − A = 14 −19 8 + 6 −10 − 9 " " ! " ! ! 2 28 −6 18 8 10 = + (c) 2A + 3B = 12 −12 24 −30 −12 18 ⎞ ⎞ ⎛ ⎛ −5 1 −2 − 3 0 + 1 ⎟ ⎟ ⎜ ⎜ 16. (a) A − B = ⎝ 4 − 0 1 − 2 ⎠ = ⎝ 4 −1 ⎠ 11 5 7+4 3+2 ⎞ ⎞ ⎛ ⎛ 5 −1 3 + 2 −1 − 0 ⎟ ⎟ ⎜ ⎜ 1⎠ 2 − 1 ⎠ = ⎝ −4 (b) B − A = ⎝ 0 − 4 −11 −5 −4 − 7 −2 − 3 ⎛ ⎞ ⎛ ⎞ 1 −1 2 −2 ⎜ ⎟ ⎜ ⎟ 3⎠ = ⎝8 (c) 2(A + B) = 2 ⎝ 4 6⎠ 3 1 6 2 " ! ! " −11 6 −2 − 9 12 − 6 = 17. (a) AB = 5 + 12 −30 + 8 17 −22 ! " ! " −2 − 30 3 + 24 −32 27 (b) BA = = 6 − 10 −9 + 8 −4 −1 " " ! ! 19 −18 4 + 15 −6 − 12 (c) A2 = = −30 31 −10 − 20 15 + 16 373 5. 3 × 4 10. Not equal 8.1 Matrix Algebra " " ! 19 6 1 + 18 −6 + 12 = 3 22 −3 + 6 18 + 4 ⎞ ⎞ ⎛ ⎛ 0 −6 5 −4 + 4 6 − 12 −3 + 8 ⎟ ⎟ ⎜ ⎜ 0 5⎠ 18. (a) AB = ⎝ −20 + 10 30 − 30 −15 + 20 ⎠ = ⎝ −10 (d) B2 = ! −32 + 12 (b) 19. (a) (b) (c) (d) 20. (a) 48 − 36 −20 −24 + 24 12 0 " ! " 2 8 −4 + 30 − 24 −16 + 60 − 36 = BA = 1 − 15 + 16 4 − 30 + 24 2 −2 ! " 9 24 BC = 3 8 "! ! " " ! 9 24 1 −2 3 8 A(BC) = = −2 4 −6 −16 3 8 ! "! " ! " 0 2 0 0 0 0 C(BA) = = 3 4 0 0 0 0 " " ! "! ! −4 −5 6 5 1 −2 = A(B + C) = 8 10 5 5 −2 4 ⎛ ⎞ 3 ⎜ ⎟ AB = ( 5 −6 7 ) ⎝ 4 ⎠ = (−16) ! −1 ⎛ ⎞ 3 ⎜ ⎟ (b) BA = ⎝ 4 ⎠ ( 5 −6 −1 ⎛ 15 −18 ⎜ (c) (BA)C = ⎝ 20 −24 −5 6 ⎛ 15 −18 ⎜ 7 ) = ⎝ 20 −24 −5 6 ⎞⎛ 1 2 21 ⎟⎜ 28 ⎠ ⎝ 0 1 −7 3 2 ⎞ 21 ⎟ 28 ⎠ −7 ⎞ ⎛ 4 78 ⎟ ⎜ −1 ⎠ = ⎝ 104 −26 1 ⎞ 54 99 ⎟ 72 132 ⎠ −18 −33 (d) Since AB is 1 × 1 and C is 3 × 3 the product (AB)C is not defined. ⎛ ⎞ 4 ⎜ ⎟ 21. (a) AT A = ( 4 8 −10 ) ⎝ 8 ⎠ = (180) −10 ⎛ ⎞ ⎛ ⎞ 2 4 8 10 ⎜ ⎟ ⎜ ⎟ (b) BT B = ⎝ 4 ⎠ ( 2 4 5 ) = ⎝ 8 16 20 ⎠ 5 10 20 25 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 6 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ (c) A + BT = ⎝ 8 ⎠ + ⎝ 4 ⎠ = ⎝ 12 ⎠ −10 5 −5 ! " ! " ! " 1 2 −2 5 −1 7 22. (a) A + BT = + = 2 4 3 7 5 11 ! " ! " " ! 2 4 −2 5 4 −1 (b) 2AT − BT = − = 4 8 1 1 3 7 4 374 8.1 " " ! "! 2 −3 −7 3 −1 = −6 −14 −3 −3 4 " "T ! ! 7 38 7 10 = (AB)T = 10 75 38 75 "! ! " " ! 3 8 5 −2 7 38 T T B A = = 10 −5 10 75 4 1 " ! " ! ! " 2 7 −3 11 5 −4 = + AT + B = −7 2 9 6 2 8 " ! " ! " ! 10 18 −3 −7 7 11 2A + BT = = + 3 14 −8 12 11 2 (c) AT (A − B) = 23. (a) (b) 24. (a) (b) 25. ! −4 8 27. ! −19 " 18 − " ! − 4 16 ! " 19 20 + " ! 1 2 ! −6 9 Matrix Algebra = ! " −38 −2 = ! −14 1 " ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 6 −5 −6 −5 ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 26. ⎝ 3 ⎠ + ⎝ −5 ⎠ + ⎝ −8 ⎠ = ⎝ −10 ⎠ 22 10 15 −3 ⎛ " −7 ⎞ ⎛ −1 ⎞ ⎛ 2 ⎞ ⎛ −10 ⎞ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 28. ⎝ 17 ⎠ + ⎝ 1 ⎠ − ⎝ 8 ⎠ = ⎝ 10 ⎠ −6 4 −6 4 29. 4 × 5 30. 3 × 2 31. AT = ! 2 4 " −3 ; 2 (AT )T = ! " " ! 6 −6 4 = A T + BT =A 32. (A + B)T = 14 10 2 " " ! "! " ! 16 −8 2 −3 4 2 −8 T T = ; B A = 40 −20 4 2 10 5 −20 2 −3 "T ! 16 40 = 33. (AB) = 40 −20 " ! 12 −18 = 6AT 34. (6A)T = 24 12 ⎞ ⎞ ⎛ ⎛ 5 15 9 2 1 ! " ⎟ 2 6 2 ⎟ ⎜ ⎜ = ⎝ 15 39 27 ⎠ = BT 35. B = AAT = ⎝ 6 3 ⎠ 1 3 5 2 5 9 27 29 T ! 16 −8 36. Using Problem 33 we have (AAT )T = (AT )T AT = AAT , so that AAT is symmetric. ! " ! " 1 0 0 0 37. Let A = and B = . Then AB = 0. 0 0 0 1 ⎛ ⎞ 2 3 4 ⎜ ⎟ 8 ⎠ = BC. 38. We see that A ̸= B, but AC = ⎝ 4 6 6 9 12 39. Since (A+B)2 = (A+B)(A+B) ̸= A2 +AB+BA+B2 , and AB ̸= BA in general, (A+B)2 ̸= A2 +2AB+B2 . 40. Since (A + B)(A − B) = A2 − AB + BA − B2 , and AB ̸= BA in general, (A + B)(A − B) ̸= A2 − B2 . 41. a11 x1 + a12 x2 = b1 ; a21 x1 + a22 x2 = b2 ⎞⎛ ⎞ ⎛ ⎞ ⎛ 7 x1 2 6 1 ⎟ ⎟⎜ ⎟ ⎜ ⎜ 42. ⎝ 1 2 −1 ⎠ ⎝ x2 ⎠ = ⎝ −1 ⎠ 5 7 −4 9 x3 375 8.1 Matrix Algebra 43. ( x ⎛ y) 0 ! a b/2 b/2 c "! " x −∂/∂z y =( ax + by/2 ∂/∂y ⎞⎛ P ⎞ bx/2 + cy ) ⎛ ! " x y =( ax2 + bxy/2 + bxy/2 + cy 2 )=( ax2 + bxy + cy 2 ) −∂Q/∂z + ∂R/∂y ⎞ ⎟ ⎟⎜ ⎟ ⎜ ⎜ 0 −∂/∂x ⎠ ⎝ Q ⎠ = ⎝ ∂P/∂z − ∂R/∂x ⎠ = curl F 44. ⎝ ∂/∂z −∂P/∂y + ∂Q/∂x R −∂/∂y ∂/∂x 0 ⎞ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎛ ⎞ ⎛ xY x cos γ + y sin γ x cos γ sin γ 0 x ⎟ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 45. (a) MY ⎝ y ⎠ = ⎝ − sin γ cos γ 0 ⎠ ⎝ y ⎠ = ⎝ −x sin γ + y cos γ ⎠ = ⎝ yY ⎠ z z zY 0 0 1 z ⎞ ⎞ ⎛ ⎛ 1 0 0 cos β 0 − sin β ⎟ ⎜ ⎟ ⎜ (b) MR = ⎝ 0 MP ⎝ 0 cos α sin α ⎠ 1 0 ⎠; 0 − sin α cos α sin β 0 cos β ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 0 0 1 1 1 0 0 1 1 √ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜1 √ 1 ⎟⎜ ⎟ 3 (c) MP ⎝ 1 ⎠ = ⎝ 0 cos 30◦ sin 30◦ ⎠ ⎝ 1 ⎠ = ⎝ 0 2 2 ⎠ ⎝ 1 ⎠ = ⎝ 2 ( 3 + 1) ⎠ √ √ 1 3 0 − sin 30◦ cos 30◦ 1 1 1 0 − 12 2 ( 3 − 1) 2 √ √ ⎞⎛ ⎞ ⎛ 2 ⎞ ⎛ ⎞ ⎛ ⎞⎛ 1 1 0 − 22 1 cos 45◦ 0 − sin 45◦ 2 ⎟⎜ √ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ 1 √ 0 ⎠ ⎝ 21 ( 3 + 1) ⎠ MR MP ⎝ 1 ⎠ = ⎝ 0 1 0 ⎠ ⎝ 2 ( 3 + 1) ⎠ = ⎝ √0 1 √ √ √ 1 1 2 2 0 1 sin 45◦ 0 cos 45◦ 2 ( 3 − 1) 2 2 2 ( 3 − 1) √ ⎞ ⎛1 √ 6) 4 (3 2 − ⎜ 1 √ ⎟ = ⎝ 2 ( 3 + 1) ⎠ √ √ 1 6) 4( 2 + √ ⎞ ⎞⎛ 1 √ ⎛ ⎞ ⎛ cos 60◦ sin 60◦ 0 6) 1 4 (3 √2 − ⎟⎜ 1 ⎜ ⎟ ⎜ ⎟ ◦ ◦ MY MR MP ⎝ 1 ⎠ = ⎝ − sin 60 cos 60 0 ⎠ ⎝ 2 ( 3 + 1) ⎠ √ √ 1 0 0 1 1 6) 4( 2 + √ √ √ √ √ ⎛ 1 √ ⎞ ⎞ ⎛ 1 ⎛ ⎞ 3 1 0 (3 2 − 6 ) (3 2 − 6 + 6 + 2 3 ) 2 2 4 8 √ √ √ ⎟ ⎜1 ⎟ ⎟⎜ 1 √ ⎜ √ 1 = ⎝ − 23 0 ⎠ ⎝ 2 ( 3 + 1) ⎠ = ⎝ 8 (−3 6 + 3 2 + 2 3 + 2) ⎠ 2 √ √ √ √ 1 1 6) 6) 0 0 1 4( 2 + 4( 2 + ! " " ! "! 1 0 2 −2 2 −2 46. (a) LU = 1 =A = 1 2 0 3 1 2 " ! "! " ! 6 2 6 2 1 0 = =A (b) LU = 2 1 4 1 0 − 31 3 ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 0 0 1 −2 1 1 −2 1 ⎜ ⎟⎜ ⎟ ⎜ ⎟ (c) LU = ⎝ 0 1 0 ⎠ ⎝ 0 1 2⎠ = ⎝0 1 2⎠ = A 2 10 1 0 0 −21 2 6 1 ⎛ ⎞ ⎛ ⎞⎛ ⎞ 1 0 0 1 1 1 1 1 1 ⎜ ⎟ ⎜ ⎟⎜ ⎟ (d) LU = ⎝ 3 1 0 ⎠ ⎝ 0 −2 −1 ⎠ = ⎝ 3 1 2⎠ = A 0 0 1 1 1 1 1 −1 1 ⎞ ⎛ 17 43 ! " " ! "! A11 A12 A11 B1 + A12 B2 B1 ⎟ ⎜ 47. (a) AB = = ⎝ 3 75 ⎠ = A21 B1 + A22 B2 B2 A21 A22 −14 51 376 8.2 Systems of Linear Algebraic Equations since A11 B1 + A12 B2 = and ! 13 25 −9 49 A21 B1 + A22 B2 = ( −24 " + ! 4 18 12 26 34 ) + ( 10 " = ! 17 43 3 75 17 ) = ( −14 " 51 ) . (b) It is easier to enter smaller strings of numbers and the chance of error is decreased. Also, if the large matrix has submatrices consisting of all zeros or diagonal matrices, these are easily entered without listing all of the entries. EXERCISES 8.2 Systems of Linear Algebraic Equations 1. ! 1 −1 4 3 11 −5 " −4R1 +R2 −−−−−−→ ! 1 −1 0 7 The solution is x1 = 4, x2 = −7. ! ! " 3 −2 4 1 −1 R12 2. −−−−− −→ 1 −1 −2 3 −2 11 −49 −2 4 " " 1 7 R2 −−−−−−→ −3R +R 2 −−−−1−−→ ! ! 1 −1 0 1 The solution is x1 = 8, x2 = 10. ) * ) " ! 1 1 13 − 59 −2R1 +R2 1 9 3 −5 9 R1 −−−−−−→ 3. −−−−−−→ 2 −1 −1 2 1 −1 0 * ) 1 0 − 23 − 1 R2 +R1 −−3−−−−→ 1 0 1 3 The solution is x1 = − 32 , x2 = ) ! " 1 1 10 15 1 10 R1 4. −−−−−−→ 3 2 −1 3 ) 1 − 32 R2 +R1 −−−−−−→ 0 11 1 −1 0 1 1 3 1 3 −7 −2 10 − 95 1 9 " " * ! R3 +R1 −−−−−−→ R +R 1 −−−2−−−→ 3R2 −−−−−−→ ! ) 4 1 0 0 1 −7 8 1 0 0 1 1 1 3 0 1 10 − 95 1 3 " " * 1 3. 3 2 1 10 2 −1 0 17 − 25 1 * 13 25 −3R1 +R2 −−−−−−→ ) 1 0 3 2 − 52 1 10 13 − 10 * − 52 R2 −−−−−−→ ) 1 3 2 0 1 1 10 13 25 * * 13 The solution is x1 = − 17 25 , x2 = 25 . ⎛ ⎞ ⎞ ⎞ ⎛ ⎛ 1 −1 −1 −3 −3 1 −1 −1 1 −1 −1 −3 1 R2 ⎜ ⎜ ⎟ −2R +R2 ⎜ ⎟ 7 13 ⎟ 3 5 5 7 13 ⎠ −−−5−−−→ ⎝ 0 1 5. ⎝ 2 7 ⎠ −−−−1−−→ ⎝0 5 5 ⎠ −R1 +R3 0 −1 4 −8 1 −2 3 −11 0 −1 4 −8 ⎛ ⎛ ⎛ ⎞ ⎞ − 52 1 0 25 1 0 52 − 52 1 0 0 2 5 R R +R − 1 ⎜ R +R1 ⎜ ⎜ 27 3 13 ⎟ 13 ⎟ 5 3 7 0 1 0 −−−2−−−→ − − − − − −→ − − − − − −→ 0 1 ⎝ 0 1 57 ⎝ ⎝ ⎠ ⎠ 5 5 5 R2 +R3 − 75 R3 +R2 27 27 0 0 1 0 0 5 −5 0 0 1 −1 The solution is x1 = 0, x2 = 4, x3 = −1. 377 0 ⎞ ⎟ 4⎠ −1 8.2 Systems of Linear Algebraic Equations ⎛ ⎛ ⎞ ⎞ 1 2 −1 0 1 2 −1 0 1 ⎟ −2R +R2 ⎜ ⎟ − 3 R2 ⎜ ⎟ 9 ⎠ −−−−1−−→ 4 9 ⎠ −−−− 1 − 34 −3 ⎠ −−→ ⎝ 0 ⎝ 0 −3 −R1 +R3 0 −3 2 3 0 −3 2 3 3 ⎛ ⎛ ⎛ ⎞ ⎞ 5 5 1 0 6 6 1 0 0 1 0 3 3 1 5 −2R2 +R1 ⎜ ⎟ − 2 R3 ⎜ ⎟ − 3 R3 +R1 ⎜ 4 4 −−−−→ ⎝ 0 1 0 −−−−−−→ ⎝ 0 1 − 3 −3 ⎠ −−−−−−→ ⎝ 0 1 − 3 −3 ⎠ −− 4 3R2 +R3 3 R3 +R2 0 0 −2 −6 0 0 1 0 0 1 3 1 2 −1 ⎜ 6. ⎝ 2 1 2 1 −1 1 0 ⎞ ⎛ The solution is x1 = 1, x2 = 1, x3 = 3. ! ! " " 1 1 1 0 −R1 +R2 1 1 1 0 7. −−−−−−→ 1 1 3 0 0 0 2 0 Since x3 = 0, setting x2 = t we obtain x1 = −t, x2 = t, x3 = 0. ! " " ! ! 1 2 −4 9 1 2 −4 1 2 −4 9 −5R1 +R2 1 − 11 R2 −−−−−−→ −−−− −−→ 8. 5 −1 2 1 0 −11 22 −44 0 1 −2 If x3 = t, the solution is x1 = 1, x2 = 4 + 2t, x3 = t ⎛ ⎞ ⎞ ⎛ 1 −1 −1 8 8 1 −1 −1 row ⎜ ⎟ ⎟ ⎜ 9. ⎝ 1 −1 1 3 ⎠ −−−−−−→ ⎝ 0 0 2 −5 ⎠ operations −1 1 1 4 0 0 0 12 Since the bottom row implies 0 = 12, the system is inconsistent. ⎛ ⎞ ⎛ 4 ⎞ 1 31 4 3 1 3 row ⎜ ⎟ ⎟ ⎜ 10. ⎝ 4 3 −3 ⎠ −−−−−−→ ⎝ 0 1 −5 ⎠ operations 2 −1 11 0 0 0 The solution is x1 = 3, x2 = −5. ⎛ ⎞ ⎛ 1 1 2 2 0 0 row ⎜ ⎟ ⎜ 11. ⎝ −2 1 1 0 ⎠ −−−−−−→ ⎝ 0 1 operations 3 0 1 0 0 0 0 0 1 3 ⎟ 0⎠ 0 1 The solution is x1 = x2 = x3 = 0. ⎛ ⎞ ⎛ 1 −1 1 −1 −2 0 row ⎜ ⎟ ⎜ 12. ⎝ 2 4 5 0 ⎠ −−−−−−→ ⎝ 0 1 operations 6 0 −3 0 0 0 The solution is ⎛ 1 2 2 ⎜ 1 1 13. ⎝ 1 1 −3 −1 x1 = 21 t, x2 = − 32 t, ⎞ ⎛ 2 1 row ⎟ ⎜ 0 ⎠ −−−−−−→ ⎝ 0 operations 0 0 ⎞ −2 0 0 0 3 2 ⎟ 0⎠ x3 = t. 2 2 1 1 2 0 4 1 The solution is x1 = −2, x2 − 2, x3 = 4. ⎛ ⎞ ⎛ 1 −2 1 1 −2 1 2 row ⎜ ⎟ ⎜ 1 14. ⎝ 3 −1 2 5 ⎠ −−−−−−→ ⎝ 0 1 −5 operations 2 1 1 1 0 0 0 ⎞ ⎞ ⎟ 2⎠ 2 ⎞ ⎟ − 15 ⎠ −2 Since the bottom row implies 0 = −2, the system is inconsistent. 378 9 4 " −2R +R 1 −−−−2−−→ ! 1 ⎞ ⎟ 1⎠ 3 1 0 0 1 0 1 −2 4 " 8.2 ⎛ 1 1 1 ⎜ 15. ⎝ 1 −1 −1 3 1 1 3 ⎞ ⎛ 3 1 1 1 row ⎟ ⎜ −1 ⎠ −−−−−−→ ⎝ 0 1 1 operations 0 0 0 5 Systems of Linear Algebraic Equations ⎞ ⎟ 2⎠ 0 If x3 = t the solution is x1 = 1, x2 = 2 − t, x3 = t. ⎛ ⎞ ⎞ ⎛ 1 −1 −2 −1 1 −1 −2 −1 row ⎜ ⎟ ⎟ ⎜ 2⎠ 1 −7 ⎠ −−−−−−→ ⎝ 0 1 1 16. ⎝ −3 −2 operations 2 3 1 0 0 0 8 0 If x3 = t the solution is x1 = 1 + t, x2 = 2 − t, x3 = t. ⎛ ⎛ 1 0 1 −1 1 ⎞ ⎞ 1 0 1 −1 1 ⎜0 1 1 ⎜0 1 3 ⎟ 3⎟ 2 1 1 row ⎜ ⎜ ⎟ 2 2 2 ⎟ 17. ⎜ ⎟ ⎟ −−−−−−→ ⎜ ⎝ 1 −1 0 ⎝ ⎠ 1 −1 operations 0 0 1 −5 1 ⎠ 2 1 1 1 1 0 0 0 1 0 The ⎛ 2 ⎜3 ⎜ 18. ⎜ ⎝1 solution is x1 = 0, x2 = 1, x3 = 1, x4 ⎛ ⎞ 1 21 3 1 1 0 ⎜ 1 1 1 4⎟ row ⎜0 1 ⎟ ⎟ −−−−−−→ ⎜ ⎠ 3 operations ⎝ 0 0 2 2 3 4 5 −2 1 16 0 0 The ⎛ 1 ⎜0 ⎜ 19. ⎜ ⎝1 = 0. 1 2 0 1 −2 1 0 3 2 1⎟ ⎟ ⎟ −1 ⎠ −1 1 0 solution is x1 = 1, x2 = 2, x3 = −1, x4 = 0. ⎛ ⎞ 3 5 −1 1 1 3 5 −1 ⎜ 0 1 1 −1 4⎟ 1 1 −1 row ⎜ ⎟ ⎟ −−−−−−→ ⎜ 2 5 −4 −2 ⎠ operations ⎝ 0 0 1 −4 6 1 4 6 −2 0 0 0 0 Since the bottom row implies 0 = 1, ⎛ ⎞ ⎛ 1 2 0 1 0 1 ⎜ 4 9 1 12 0 ⎟ ⎜0 row ⎜ ⎟ ⎜ 20. ⎜ ⎟ −−−−−−→ ⎜ ⎝ 3 9 6 21 0 ⎠ operations ⎝ 0 1 3 1 9 0 0 ⎞ 1 ⎞ 4⎟ ⎟ ⎟ 1⎠ 1 the system is inconsistent. ⎞ 2 0 1 0 1 1 8 0⎟ ⎟ ⎟ 0 1 −2 0 ⎠ 0 0 0 If x4 = t the solution is x1 = 19t, x2 = −10t, x3 ⎞ ⎛ ⎛ 4.280 1 1 1 1 row ⎜ ⎟ ⎜ 21. ⎝ 0.2 −0.1 −0.5 −1.978 ⎠ −−−−−−→ ⎝ 0 operations 1.686 4.1 0.3 0.12 0 The solution is x1 = 0.3, x2 = −0.12, x3 = 4.1. ⎛ ⎞ ⎛ 2.6170 1 2.5 1.4 4.5 row ⎜ ⎟ ⎜ 1.2 0.7545 ⎠ −−−−−−→ ⎝ 0 22. ⎝ 1.35 0.95 operations 2.7 3.05 −1.44 −1.4292 0 0 = 2t, x4 = t. 1 1 1 2.333 0 1 0.56 1 0 4.28 ⎟ 9.447 ⎠ 4.1 1.8 −6.3402 The solution is x1 = 1.45, x2 = −1.62, x3 = 0.28. ⎞ 1 1.0468 ⎞ ⎟ −3.3953 ⎠ 0.28 23. From x1 Na + x2 H2 O → x3 NaOH + x4 H2 we obtain the system x1 = x3 , 2x2 = x3 + 2x4 , x2 = x3 . We see that x1 = x2 = x3 , so the second equation becomes 2x1 = x1 + 2x4 or x1 = 2x4 . A solution of the system is x1 = x2 = x3 = 2t, x4 = t. Letting t = 1 we obtain the balanced equation 2Na + 2H2 O → 2NaOH + H2 . 379 8.2 Systems of Linear Algebraic Equations 24. From x1 KClO3 → x2 KCl + x3 O2 we obtain the system x1 = x2 , x1 = x2 , 3x1 = 2x3 . Letting x3 = t we see that a solution of the system is x1 = x2 = 32 t, x3 = t. Taking t = 3 we obtain the balanced equation 2KClO3 → 2KCl + 3O2 . 25. From x1 Fe3 O4 + x2 C → x3 Fe + x4 CO we obtain the system 3x1 = x3 , 4x1 = x4 , x2 = x4 . Letting x1 = t we see that x3 = 3t and x2 = x4 = 4t. Taking t = 1 we obtain the balanced equation Fe3 O4 + 4C → 3Fe + 4CO. 26. From x1 C5 H8 + x2 O2 → x3 CO2 + x4 H2 O we obtain the system 5x1 = x3 , 8x1 = 2x4 , 2x2 = 2x3 + x4 . Letting x1 = t we see that x3 = 5t, x4 = 4t, and x2 = 7t. Taking t = 1 we obtain the balanced equation C5 H8 + 7O2 → 5CO2 + 4H2 O. 27. From x1 Cu + x2 HNO3 → x3 Cu(NO3 )2 + x4 H2 O + x5 NO we obtain the system x1 = 3, x2 = 2x4 , x2 = 2x3 + x5 , 3x2 = 6x3 + x4 + x5 . Letting x4 = t we see that x2 = 2t and 2t = 2x3 + x5 6t = 6x3 + t + x5 Then x3 = 3 4t and x5 = 1 2 t. Finally, x1 = x3 = 3 4 t. 2x3 + x5 = 2t or 6x3 + x5 = 5t. Taking t = 4 we obtain the balanced equation 3Cu + 8HNO3 → 3Cu(NO3 )2 + 4H2 O + 2NO. 28. From x1 Ca3 (PO4 )2 + x2 H3 PO4 → x3 Ca(H2 PO4 )2 we obtain the system 3x1 = x3 , 2x1 + x2 = 2x3 , 8x1 + 4x2 = 8x3 , 3x2 = 4x3 . Letting x1 = t we see from the first equation that x3 = 3t and from the fourth equation that x2 = 4t. These choices also satisfy the second and third equations. Taking t = 1 we obtain the balanced equation Ca3 (PO4 )2 + 4H3 PO4 → 3Ca(H2 PO4 )2 . 29. The system of equations is −i1 + i2 − i3 = 0 10 − 3i1 + 5i3 = 0 −i1 + i2 − i3 = 0 or 3i1 − 5i3 = 10 27 − 6i2 − 5i3 = 0 Gaussian elimination gives ⎛ −1 1 −1 ⎜ ⎝ 3 0 −5 0 6 5 0 6i2 + 5i3 = 27 ⎛ ⎞ 1 −1 row ⎜ ⎟ 1 10 ⎠ −−−−−−→ ⎝ 0 operations 0 0 27 35 38 1 , i2 = , i3 = . 9 9 3 30. The system of equations is i1 − i2 − i3 = 0 1 −8/3 1 0 ⎟ 10/3 ⎠ . 1/3 The solution is i1 = 52 − i1 − 5i2 = 0 i1 − i2 − i3 = 0 or i1 + 5i2 = 52 5i2 − 10i3 = 0 −10i3 + 5i2 = 0 380 ⎞ 8.2 Gaussian elimination gives ⎛ 1 −1 ⎜ ⎝1 0 5 5 −1 0 ⎞ Systems of Linear Algebraic Equations ⎛ 1 −1 −1 row ⎟ ⎜ 52 ⎠ −−−−−−→ ⎝ 0 1 1/6 operations 0 0 1 0 0 −10 0 ⎞ ⎟ 26/3 ⎠ . 4 The solution is i1 = 12, i2 = 8, i3 = 4. 31. Interchange row 1 and row in I3 . 32. Multiply row 3 by c in I3 . 33. Add c times row 2 to row 3 in I3 . 34. Add row 4 to row 1 in I4 . ⎛ a21 ⎜ 35. EA = ⎝ a11 a31 ⎛ ⎜ 37. EA = ⎝ a22 a12 a32 a11 ⎜ 38. E1 E2 A = E1 ⎝ a11 ⎜ 36. EA = ⎝ a21 ca31 a33 ⎞ a12 a22 a13 a23 ca22 + a32 ca23 + a33 a21 ca21 + a31 ⎛ ⎛ ⎞ a23 ⎟ a13 ⎠ a11 a12 a21 ca21 + a31 a22 ca22 + a32 1 0 1 2 1 Y= we have This implies y1 = 2 and a21 ! "! ! y1 y2 1 0 1 2 1 2 −2 0 3 " " 2 −2 ! = "! y1 y2 X= 0 " 3 a12 ca22 + a32 a23 ⎞ ⎟ a13 ⎠ ca23 + a33 ! " 2 . 6 " X ! " 2 = . 6 5 3 and 2x1 − 2x2 = 2x1 − 10 3 8 3 = 2 or x1 = 40. The system is equivalent to ! 1 0 2 3 1 "! Y= we have ⎞ a13 ⎟ a23 ⎠ ca33 + y2 = 1 + y2 = 6 or y2 = 5. Then ! "! " ! " 2 2 −2 x1 = , 5 0 3 x2 which implies 3x2 = 5 or x2 = Letting ca32 a22 ⎟ ⎜ a11 a23 ⎠=⎝ ca21 + a31 ca23 + a33 ! 1 2 y1 ⎛ ⎞ a13 39. The system is equivalent to Letting ⎟ ⎠ a12 a22 ! ! 6 2 0 − 31 y1 y2 1 0 2 3 1 " "! = y1 y2 " X= ! 6 0 " = 381 ! 2 − 13 ! 1 −1 " 1 −1 X " . " . . The solution is X = 5 3, 3 +8 , . 8.2 Systems of Linear Algebraic Equations This implies y1 = 1 and 32 y1 + y2 = 2 3 + y2 = −1 or y2 = − 53 . Then "! " ! " ! x1 1 6 2 , = − 35 x2 0 − 31 + , which implies − 31 x2 = − 35 or x2 = 5 and 6x1 + 2x2 = 6x1 + 10 = 1 or x1 = − 32 . The solution is X = − 32 , 5 . 41. The system is equivalent to ⎛ Letting ⎞⎛ ⎛ ⎞ ⎞ 0 1 −2 1 2 ⎟⎜ ⎜ ⎟ ⎟ 0⎠⎝0 1 2 ⎠ X = ⎝ −1 ⎠ . 1 0 0 −21 1 1 0 ⎜ ⎝0 1 2 10 ⎛ y1 ⎞ 1 ⎟ 2⎠X ⎛ ⎞ 1 −2 ⎜ ⎟ ⎜ Y = ⎝ y2 ⎠ = ⎝ 0 1 0 y3 we have ⎛ 1 0 ⎜ 1 ⎝0 2 10 0 −21 ⎞ 2 ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠ ⎝ y2 ⎠ = ⎝ −1 ⎠ . 1 1 y3 0 ⎞⎛ y1 ⎛ ⎞ This implies y1 = 2, y2 = −1, and 2y1 + 10y2 + y3 = 4 − 10 + y3 = 1 or y3 = 7. Then ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 −2 1 x1 2 ⎜ ⎟⎜ ⎟ ⎜ ⎟ 1 2 ⎠ ⎝ x2 ⎠ = ⎝ −1 ⎠ , ⎝0 0 0 −21 7 x3 which implies −21x3 = 7 or x3 = − 13 , x2 + 2x3 = x2 − 32 = −1 or x2 = − 13 , and x1 − 2x2 + x3 = x1 + 32 − 13 = 2 , + or x1 = 35 . The solution is X = 53 , − 13 , − 31 . 42. The system is equivalent to ⎛ Letting 1 0 ⎜ ⎝3 1 1 1 ⎛ ⎞ 0 ⎜ ⎟ ⎟ ⎟⎜ 0 ⎠ ⎝ 0 −2 −1 ⎠ X = ⎝ 1 ⎠ . 0 0 1 1 4 0 ⎞⎛ ⎛ we have y1 1 ⎞ 1 ⎛ 1 1 ⎞ 1 1 ⎞ ⎜ ⎟ ⎜ ⎟ Y = ⎝ y2 ⎠ = ⎝ 0 −2 −1 ⎠ X 0 0 1 y3 ⎛ 1 0 ⎜ ⎝3 1 1 1 ⎞⎛ ⎛ ⎞ 0 ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠ ⎝ y2 ⎠ = ⎝ 1 ⎠ . 1 4 y3 0 y1 ⎞ This implies y1 = 0, 3y1 + y2 = y2 = 1, and y1 + y2 + y3 = 0 + 1 + y3 = 4 or y3 = 3. Then ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 1 1 1 x1 ⎜ ⎟⎜ ⎟ ⎜ ⎟ = 0 −2 −1 x ⎝ ⎠⎝ 2 ⎠ ⎝1⎠, 3 0 0 1 x3 which implies x3 = 3, −2x2 − x3 = −2x2 − 3 = 1 or x2 = −2, and x1 + x2 + x3 = x1 − 2 + 3 = 0 or x1 = −1. The solution is X = (−1, −2, 3). 382 8.3 Rank of a Matrix 43. Using the Solve function in Mathematica we find x1 = −0.0717393 − 1.43084c, x2 = −0.332591 + 0.855709c, x3 = c, where c is any real number 44. Using the Solve function in Mathematica we find x1 = c/3, x2 = 5c/6, x3 = c, where c is any real number 45. Using the Solve function in Mathematica we find x1 = −3.76993, x2 = −1.09071, x3 = −4.50461, x4 = −3.12221 46. Using the Solve function in Mathematica we find x1 = where b and c are any real numbers. 8 3 − 37 b + 32 c, x2 = EXERCISES 8.3 Rank of a Matrix 1. ! 2. ! 3 −1 " row −−−−−−→ operations ! 1 3 0 1 2 −2 " row −−−−−−→ operations ! 1 −1 1 3 0 0 2 1 ⎜ 3. ⎝ 6 3 −1 − 12 1 1 2 ⎞ ⎛ 1 1 ! 3 6 −1 2 0 2 4 5 ⎛ 1 −2 1 4 ⎛ 1 ⎜1 ⎜ 8. ⎜ ⎝0 2 ; ⎛ 1 ⎞ ⎛ 1 The rank is 1. 1 2 3 2 0 0 ⎟ 0 ⎠; 0 The rank is 3. ⎞ The rank is 3. 2 1 ⎟ 5 ⎠; 1 " row −−−−−−→ operations ⎛ ⎞ 1 1 3 4 ⎞ 1 0 1 −2 ⎜0 −6 ⎟ row ⎜ ⎟ ⎟ −−−−−−→ ⎜ −1 ⎠ operations ⎝ 0 0 5 −2 ) ⎞ 1⎟ ⎟ ⎟; 0⎠ 0 ⎛1 ⎜0 4 6 8⎟ row ⎟ ⎜ ⎟ −−−−−−→ ⎜ 1 0 0 ⎠ operations ⎝ 0 5 6 8 0 The rank is 1. ⎞ 1 0 ⎞ row ⎟ ⎟ ⎜ 4 ⎠ −−−−−−→ ⎝ 0 1 −3 ⎠; operations 1 0 0 1 ⎜ 5. ⎝ 1 0 1 4 ⎜3 ⎜ 7. ⎜ ⎝7 The rank is 2. " row ⎟ ⎜ 4 ⎠ −−−−−−→ ⎝ 0 operations 3 0 ⎜ 4. ⎝ −1 2 −1 0 6. 0 ; ⎞ ⎛ 1 3 row ⎟ ⎜ 9 ⎠ −−−−−−→ ⎝ 0 operations − 32 0 ⎛ ⎛ 0 " − 31 1 2 3 0 0 5 4 * ; The rank is 2. The rank is 2. −2 3 1 0 0 1 0 0 4⎞ 0⎟ ⎟ ; 4 ⎟ ⎠ 3 The rank is 3. 0 383 2 3 − 31 b − 31 c, x3 = −3, x4 = b, x5 = c, 8.3 Rank of a Matrix ⎛0 2 4 2 1 0 5 1 2 3 3 6 6 6 12 ⎜4 ⎜ 9. ⎜ ⎝2 1 ⎜0 ⎜ ⎜ 10. ⎜ 0 ⎜ ⎝0 ⎛ 0 0 1 2 1 1 3 4 3 3 2 1 0 0 1 0 0 0 1 6 1 −3 ⎞ ⎟ ⎟ ⎟; 2⎠ The rank is 3. 0 ⎞ ⎛ ⎞ 1 −2 1 8 −1 1 1 6 1 6 ⎜0 0 1 3 −1 1 1 5 ⎟ 1 5⎟ ⎟ ⎜ ⎟ row ⎟ ⎜ ⎟ 0 0 0 0 1 9 3 ⎟; 1 3 −1 2 10 8 ⎟ −−−−−−→ ⎜ 0 ⎟ ⎟ operations ⎜ ⎝0 0 0 0 0 0 1 0⎠ 0 0 0 1 1 3⎠ 0 0 0 0 0 0 0 0 1 8 −1 1 2 6 ⎞ ⎞ ⎛ 1 2 3 3 row ⎟ ⎟ ⎜ 1 ⎠ −−−−−−→ ⎝ 0 1 1 ⎠; operations 0 0 1 5 −2 1 8 −1 1 0 1 3 −1 1 0 0 1 −2 1 ⎜ 11. ⎝ 1 1 2 0 −1 ⎛ ⎛1 2⎞ ⎜0 1⎟ row ⎜ ⎟ −−−−−−→ ⎜ 1 ⎟ ⎠ operations ⎝ 0 3 The rank is 4. Since the rank of the matrix is 3 and there are 3 vectors, the vectors are linearly independent. ⎛ 2 ⎜1 ⎜ 12. ⎜ ⎝3 2 ⎛ 1 −1 4⎞ ⎜0 4⎟ 1 − 58 ⎟ row ⎟ ⎜ ⎟ ⎟ −−−−−−→ ⎜ ⎟ 1 ⎠ operations ⎝ 0 0 1⎠ 4 0 0 0 rank of the matrix is 3 and there are 4 vectors, the vectors are linearly dependent. ⎞ ⎞ ⎛ 3 −1 1 −1 3 −1 row ⎟ ⎟ ⎜ 4 2 ⎠ −−−−−−→ ⎝ 0 0 1 3⎠ operations 5 7 0 0 0 1 6 3 −1 2 5 Since the ⎛ 1 −1 ⎜ 13. ⎝ 1 −1 1 −1 Since the ⎛ 2 1 ⎜2 2 ⎜ 14. ⎜ ⎝ 3 −1 ⎞ rank of the matrix is 3 and there are 3 vectors, the vectors are linearly independent. ⎞ ⎛ ⎞ 1 5 1 1 1 −1 ⎜ 0 1 1 −7 ⎟ 1 1⎟ row ⎟ ⎜ ⎟ ⎟ ⎟ −−−−−−→ ⎜ 6 1 ⎠ operations ⎝ 0 0 1 −3 ⎠ 0 0 0 1 1 1 1 −1 Since the rank of the matrix is 4 and there are 4 vectors, the vectors are linearly independent. 15. Since the number of unknowns is n = 8 and the rank of the coefficient matrix is r = 3, the solution of the system has n − r = 5 parameters. 16. (a) The maximum possible rank of A is the number of rows in A, which is 4. (b) The system is inconsistent if rank(A) < rank(A/B) = 2 and consistent if rank(A) = rank(A/B) = 2. (c) The system has n = 6 unknowns and the rank of A is r = 3, so the solution of the system has n − r = 3 parameters. 17. Since 2v1 + 3v2 − v3 = 0 we conclude that v1 , v2 , and v3 are linearly dependent. Thus, the rank of A is at most 2. 18. Since the rank of A is r = 3 and the number of equations is n = 6, the solution of the system has n − r = 3 parameters. Thus, the solution of the system is not unique. 19. The system consists of 4 equations, so the rank of the coefficient matrix is at most 4, and the maximum number of linearly independent rows is 4. However, the maximum number of linearly independent columns is the same 384 8.4 Determinants as the maximum number of linearly independent rows. Thus, the coefficient matrix has at most 4 linearly independent columns. Since there are 5 column vectors, they must be linearly dependent. 20. Using the RowReduce in Mathematica we find ⎛ 1 0 0 ⎜ ⎜0 1 0 ⎜ ⎜ ⎜0 0 1 ⎜ ⎜ ⎝0 0 0 0 0 0 that the reduced row-echelon form of the augmented matrix is 834 261 ⎞ 0 0 − 443 2215 1818 282 ⎟ 0 0 2215 443 ⎟ ⎟ 13 6 ⎟ . 0 0 − 443 443 ⎟ ⎟ ⎟ 130 4214 − 443 1 0 ⎠ 2215 0 − 6079 2215 1 677 443 226 We conclude that the system is consistent and the solution is x1 = − 443 − 6 − x3 = − 443 13 443 c, 130 x4 = − 443 − 4214 2215 c, x5 = 677 433 + 6079 2215 c, 834 2215 c, x2 = 282 443 − x6 = c. EXERCISES 8.4 Determinants 2 -=9 5- - 1 1. M12 = -−2 - 1 −1 -=1 3. C13 = (−1)1+3 -−2 35. M33 7. C34 9. −7 -0 = -1 -1 2 2 1 0 -3- = 2 2- -0 2 = (−1)3+4 - 1 2 -1 1 4 -−2 - = 10 110. 2 4 -2- - 2 4. C22 = (−1)2+2 -−2 6. M41 13. (1 − λ)(2 − λ) − 6 = λ2 − 3λ − 4 -0 15. - 3 -0 -3 17. - 2 -2 -2 2. M32 = -1 8. C23 11. 17 -2 = -2 -1 4 -= 18 5- 0 -−2 3 - = 24 0 −1 4 -0 2 = (−1)2+3 - 5 1 -1 1 0 -−1 - = 22 2- 12. −1/2 14. (−3 − λ)(5 − λ) − 8 = λ2 − 2λ − 23 0 --2 0- = −48 16. 1 - = −3 -5 880 2 --7 1-2 77 1- = 3- + 2 - 2 6 - = 3(22) + 2(−2) = 62 6 4 6 42 0 5 385 -5 -0 -0 0 0 -- −3 −3 0 - = 5 -0 0 2- 0 -= 5(−3)(2) = −30 2- 1818 2215 c, 8.4 18. 19. 20. 21. 22. 23. 24. Determinants - 1 −1 −1 - - - - 2 −2 -- −1 −1 - - −1 −1 - 2 2 −2 = − 2 + - - 1 - - 2 −2 - = 20 − 2(−8) + 4 = 40 1 9 9 -1 1 9-4 5 3-1 3-2 3-1 2-=0 − 5+ 3-1 2 3- = 42 31 31 2-1 2 3-1 - 4 6 0-1 - 3 8 0 - = 0, expanding along the third column. - 1 9 02 - −2 −1 4 - −1 4 -6 1- −1 4 - − 3- + 36 1 - = −2 -- −3 - 6 1 - = −2(44) + 3(−24) − 3(−25) = −85 - 4 84 8- −3 4 8 - 3 5 1 -- - 2 5- −1 5 - - −1 2 -= 3(40) − 5(−45) + (−10) = 335 2 5- = 3+ − 5- −1 −4 10 7 10 - - 7 −4 - 7 −4 10 -1 1 1- - - y z -- -- x z -- -- x y -- = (4y − 3z) − (4x − 2z) + (3x − 2y) = −x + 2y − z -+-−- x y z - = -3 4- -2 4- -2 3-2 3 4- 1 1 1 -- - - y -z -- -- x z -- -- x - -- y + − y z -=- x 3 + y 4 + z- -2 + x 4 + z- -2 + x 3 + y-2 + x 3 + y 4 + z- = (4y + yz − 3z − yz) − (4x + xz − 2z − xz) + (3x + xy − 2y − xy) = −x + 2y − z -1 1 −3 0 -1 1 −3 --1 - 1 −3 - 1 −3 5 3 2- = 8(−5) − 16(4) = −104 − 2(8) 1 - = 2(4) 25. - = 2 - 1 −2 - 1 −2 1 1−2 11 0 -4 8 0-4 8 0 0-2 1 −2 1 -2 - 2 −2 1 1 −2 --0 5 0 46 1 - = 5(0) + 4(80) = 320 1 0- + 4-1 26. - = 5-1 -1 6 1 0- 5 −1 -5 11 1- 5 −1 1 1 27. Expanding along the first column in the original matrix and each succeeding minor, we obtain 3(1)(2)(4)(2) = 48. 28. Expanding along the bottom row we obtain - -2 0 0 -0 −2 -- -- 2 2 0 -1 6 0 -0 5 -- -- 1 1 6 −1 - = −1(−48) + 0 = 48. -+- 1 2 −1 1 -- -- 1 0 2 −1 -- 2 1 −2 3 - - 2 0 1 −2 - 29. Solving λ2 − 2λ − 15 − 20 = λ2 − 2λ − 35 = (λ − 7)(λ + 5) = 0 we obtain λ = 7 and −5. 30. Solving −λ3 + 3λ2 − 2λ = −λ(λ − 2)(λ − 1) = 0 we obtain λ = 0, 1, and 2. 386 8.5 Properties of Determinants EXERCISES 8.5 Properties of Determinants 1. Theorem 8.11 2. Theorem 8.14 3. Theorem 8.14 4. Theorem 8.12 and 8.11 5. Theorem 8.12 (twice) 6. Theorem 8.11 (twice) 7. Theorem 8.10 8. Theorem 8.12 and 8.9 9. Theorem 8.8 10. Theorem 8.11 (twice) 11. det A = −5 12. det B = 2(3)(5) = 30 13. det C = −5 14. det D = 5 15. det A = 6( 32 )(−4)(−5) = 80 16. det B = −a13 a22 a31 17. det C = (−5)(7)(3) = −105 18. det D = 4(7)(−2) = −56 19. det A = 14 = det AT 20. det A = 96 = det T - 0 −2 2 7 23 - = −80 = 20(−4) = det A det B 21. det AB = - 10 - 8 4 16 22. From Problem 21, (det A)2 = det A2 = det I = 1, -a 23. Using Theorems 8.14, 8.12, and 8.9, det A = - b -c 24. Using Theorems 8.14 and 8.9, 1 x det A = -x + y + z - -1 1 5 -- -- 1 - 3 6- = -0 25. - 4 - - 0 −1 1 - - 0 - -2 4 5 -- -- 2 - 26. - 4 2 0- = -0 - - 8 7 −2 - - 0 - 1 5 -- -- 1 - −1 −14 - = - 0 - −1 1- -0 1 y x+y+z so det A = ±1. -a 1 1 2 -1 2- = 2- b 1 -c 1 1 2- 1 -1 - = 0. 1- -1 1 z - = (x + y + z) - x -1 x+y+z- 1 5 -−1 −14 - = 1(−1)(15) = −15 0 15 - -2 -2 4 5 -4 5 -3 5 - = −2 - 0 −6 −10 - = −2 - 0 -0 - 0 −9 −22 −9 −22 - 387 4 3 0 1 y 1 1 -z - = 0. 1- 5 -5 - = −2(2)(3)(−7) = 84 −7 - 8.5 Properties of Determinants - - - −1 2 3 -- -- −1 2 3 -- -- −1 2 - - 27. - 4 −5 −2 - = - 0 3 10 - = - 0 3 - - - 9 −9 6 - - 0 9 33 - - 0 0 - −2 - 1 −2 -1 2 −6 -2 -28. - 5 0 1- = −- 5 0 1- = −-0 - 1 −2 - −2 -0 22 −6 - 3 -10 - = −1(3)(3) = −9 3−2 10 −2 -1 2 -−9 - = − - 0 -0 −2 - −2 10 0 19 - = −1(10)(− ) = 38 5 − 19 2 −9 5 - - - 1 −2 2 1 - - 1 −2 2 1 -- - 1 −2 2 1- -2 1 −2 3 -- -- 0 5 −6 1 -- -- 0 5 −6 1 -29. -=-=- = 1(5)(−2)(0) = 0 -3 4 −8 1 - - 0 10 −14 −2 - - 0 0 −2 −4 - - - 3 −11 12 2 - - 0 −5 6 −1 - - 0 0 0 0-1 2 -1 2 -1 -1 2 2 0-0 1 4 52 2 0 -2 2 0 --0 -2 5 0 1-2 5 0 1−4 −4 1 1 −4 1 -0 1 -0 1 30. - = −- = −- = −- = −-0 0 -0 0 -0 -0 1 4 5-1 2 2 08 8 4 -1 4 5 -- 0 0 −23 - 0 0 −23 7 - 0 −5 −3 2 -3 1 3 2-3 1 3 237 ) = −148 2 - 3 4- -1 - 2 3 -- -- 0 -=0 −1 - - 0 - 5 16 - - 0 = −(1)(1)(8)( -1 2 -1 3 31. -2 3 -1 5 -2 9 -1 3 32. -0 1 -3 1 3 5 6 8 1 7 6 4 - 4- -1 2 - 7- -0 1 -=7 - - 0 −1 - 20 - - 0 3 -1 3 7 8-2 9 1 4- = −-0 1 6 5-3 1 4 2 -1 3 7 -0 1 6 =- 0 0 −31 -0 0 31 -1 4-0 8- = −-0 5-0 2 - 4- -1 - 5 -- -- 0 -=−15 - - 0 - 30 - - 0 2 - 4- -1 2 3 4- 1 2 3 -- -- 0 1 2 3 --=- = 1(1)(2)(8) = 16 0 2 2- -0 0 2 2- 0 −1 7 - - 0 0 0 8 - 3 7 4- -1 3 7 4 -- 3 −13 0 -- -- 0 1 6 5 --=1 6 5- -0 3 −13 0- −8 −17 −10 - - 0 −8 −17 −10 3 7 41 6 5 -- = 1(1)(−31)(15) = −465 0 −31 −15 0 0 15 - 0 1 4 37 2 3 33. We first use the second row to reduce the third row. Then we use the first row to reduce - -1 1 -1 1 1 1 1 -- -- 1 - b−a c − a - = (b − a)(c − a) - 0 1 b c - = -0 -a -0 b - 0 b2 − ab c2 − ac - - 0 b(b − a) c(c − a) Expanding along the first row gives (b − a)(c − a)(c − b). the second row. 1 -1-. c- 34. In order, we use the third row to reduce the fourth row, the second row to reduce the third row, and the first row to reduce the second row. We then pull out a common factor from each column. - -1 1 1 1 - 1 1 1 1 - -1 1 1 1 - -0 1 1 1 - a b c d - -0 b−a c−a d − a -- - = (b − a)(c − a)(d − a) -=-. - 2 2 2 -0 b c d - a b c d2 - - 0 b2 − ab c2 − ac d2 − ac - - 0 b2 c2 d2 - a3 b3 c3 d3 - - 0 b3 − ab2 c3 − ac2 d3 − ad2 Expanding along the first column and using Problem 33 we obtain (b − a)(c − a)(d − a)(c − b)(d − b)(d − c). 388 8.6 Inverse of a Matrix 35. Since C11 = 4, C12 = 5, and C13 = −6, we have a21 C11 + a22 C12 + a23 C13 = (−1)(4) + 2(5) + 1(−6) = 0. Since C12 = 5, C22 = −7, and C23 = −3, we have a13 C12 + a23 C22 + a33 C32 = 2(5) + 1(−7) + 1(−3) = 0. 36. Since C11 + −7, C12 = −8, and C13 = −10 we have a21 C11 + a22 C12 + a23 C13 = −2(−7) + 3(−8) − 1(−10) = 0. Since C12 = −8, C22 = −19, and C32 = −7 we have a13 C12 + a23 C22 + a33 C32 = 5(−8) − 1(−19) − 3(−7) = 0. - 10 0 -= −30; det A + det B = 10 − 31 = −21 37. det(A + B) = 0 −3 38. det(2A) = 25 det A = 32(−7) = −224 39. Factoring −1 out of each row we see that det(−A) = (−1)5 det A = − det A. Then − det A = det(−A) = det AT = det A and det A = 0. 40. (a) Cofactors: 25! ≈ 1.55(1025 ); Row reduction: 253 /3 ≈ 5.2(103 ) (b) Cofactors: about 90 billion centuries; Row reduction: about 1 10 second EXERCISES 8.6 Inverse of a Matrix 1. AB = ! ⎛ ⎜ 2. AB = ⎝ 3 − 2 −1 + 1 6 − 6 −2 + 3 2−1 " = ! 1 0 0 1 −1 + 1 " 6−6 −3 + 4 2 + 1 − 3 −1 − 1 + 2 ⎛ ⎞ 0 0 ⎟ ⎜ 6 − 6 ⎠ = ⎝0 2+2−3 0 1 ⎟ 0⎠ −1 1 = 9 ! 1 −4 1 5 " 4. det A = 5. A is nonsingular. A−1 1 = 5 ! 3 −4 1 " 2 3 0 6 3. det A = 9. A is nonsingular. A 5. det A = 12. A is nonsingular. A 2 ⎞ 1 −2 + 2 −1 6. det A = −3π . A is nonsingular. A 1 = 12 −1 7. det A = −16. A is nonsingular. A−1 8. det A = 0. A is singular. 9. det A = −30. A is nonsingular. A−1 ! 1 3 1 =− 2 3π ⎛ = " ! 1 1 9 − 94 1 9 5 9 3 5 4 −5 1 5 1 15 = ) = )1 π π 8 1 ⎜ =− ⎝ 2 16 −6 ⎛ 0 ) −14 1 ⎜ = − ⎝ −2 30 −4 6 1 4 π −2π −8 −4 4 0 1 2 " * * * = ) 1 − 3π 1 − 3π ⎞ ⎛ 1 −2 −8 ⎟ ⎜ 1 6 ⎠ = ⎝ −8 −2 1 − 3π 3 8 ⎞ ⎛ 7 13 16 15 ⎟ ⎜ 1 4 −2 ⎠ = ⎝ 15 2 −7 −4 15 389 2 3π * 1 2 1 4 − 14 13 − 30 2 − 15 7 30 1 2 − 83 1 8 ⎞ ⎟ ⎠ 8 − 15 1 15 2 15 ⎞ ⎟ ⎠ 8.6 Inverse of a Matrix ⎞⎛ 4 10 8 20 2 39 39 1 ⎜ ⎟⎜ 1 −1 5 10. det A = 78. A is nonsingular. A = ⎝ −2 −5 19 ⎠ ⎝ − 39 − 78 78 3 2 − 26 12 −9 3 13 ⎞ ⎛1 ⎛ 0 −12 0 0 3 1 ⎜ ⎟ ⎜ −1 11. det A = −36. A is nonsingular. A = − ⎝ 0 −6 0 ⎠ = ⎝ 0 61 36 0 0 0 0 18 ⎞ ⎞ ⎛ ⎛ 0 0 18 0 0 2 1 ⎜ ⎟ ⎜ ⎟ 12. det A = 16. A is nonsingular. A−1 = ⎝ 8 0 0 ⎠ = ⎝ 21 0 0 ⎠ 16 0 16 0 0 1 0 ⎛ 4 −4 14. det A = −6. A is nonsingular. A−1 1 0 0 1 ! 6 0 −2 4 16. ! 8 0 1 0 0 1 2 0 1 17. ! 1 3 1 0 15. A 18. 5 3 −1 ! 5 12 2 −3 −2 A−1 ⎛ = 0 1 ) 1 −4 1 " " " 2 3 1 1 4 1 − 12 0 0 ⎛ −1 4 3 0 2 3 1 0 ⎜ 21. ⎝ 2 1 0 −1 −2 0 0 " 1 6 0 1 4 0 ! 1 0 0 1 1 8 0 0 2 ! 1 0 2R2 −−−−−−→ 1 − 31 1 R1 −5R1 +R2 ! 0 ⎜ 1⎜ 0 =− ⎜ 6⎝ 0 −6 ⎛ 0 4 R2 1 ⎜ 19. ⎝ 4 5 6 0 1 7 8 9 0 0 ⎛ 1 0 −1 1 ⎜ 20. ⎝ 0 −2 1 0 2 −−−1−−−→ −−−8−−−→ 4 0 1 " ! 2 23 = 1 1 1 1 6 R1 3 −12 " " 0 −5 1 3 0 −−−−−−→ ! !1 1 R2 − 12 15 − 21 0 1 6 0 1 0 8 0 0 2 −−−−−−→ ! 1 12 1 4 " − 13 7 9 1 27 17 27 4 − 27 ; A 2 9 − 29 1 9 − 12 1 2 − 52 3 2 = ⎞ ⎟ ⎟ ⎟ 0⎠ 0 −1 − 34 ⎞ − 19 ⎟ ⎟ ⎟ − 17 ⎠ 9 4 9 1 2 − 12 3 1 6 1 1 ⎞ ⎟ 0⎠ 27 4 27 12 −1 −3 1 3 R2 +R1 " ⎟ ⎠ ⎞ ⎛ 0 − 61 1 −3 3 ⎜ 1 3 −9 ⎟ ⎟ ⎜ 0 − 61 ⎟=⎜ 1 −2 0 0⎠ ⎝0 ; A−1 = 1 ⎞ ⎞ ⎛ 2 21 −9 −36 9 ⎜− 1 1 6 −3 ⎟ ⎟ ⎜ 27 ⎟=⎜ 17 −6 −51 ⎠ ⎝ 10 6 ⎜ 1 ⎜ −1 = ⎜ 27 ⎝ 10 ⎛ 13. det A = 27. A is nonsingular. A−1 1 39 19 78 1 26 1 12 1 4 )1 6 0 * " 1 3 0 1 1 0 5 12 1 − 12 " −3R2 +R1 −−−−−−→ ) 1 0 0 1 1 4 1 − 12 − 14 5 12 * 1 R1 −−−2−−−→ 0 ⎞ ! − 23 1 −2 4 ⎛ 1 2 row ⎟ ⎜ 0 ⎠ −−−−−−→ ⎝ 0 1 operations 0 0 1 ⎞ ⎛ 1 0 0 row ⎟ ⎜ 1 0 ⎠ −−−−−−→ ⎝ 0 operations 0 1 0 ⎞ ⎛ 0 0 −1 R13 ⎜ ⎟ 1 0 ⎠ −−−−−−→ ⎝ 2 0 1 4 1 2 0 0 1 " 2R +R 1 2 −−−− −−→ 1 3 2 0 0 0 − 13 4 3 ! 1 0 − 23 1 1 2 0 1 1 " 3 R2 +R1 2 −− −−−−→ − 94 1 −2 0 1 0 2 3 − 19 0 0 0 1 1 0 390 1 0 2 3 2 0 1 1 1 2 9 1 9 2 9 ⎞ " ⎞ ⎟ 0 ⎠; A is singular. 1 −2 1 ⎛ 5 ⎞ 5 − 19 29 0 0 9 9 ⎜ 2 5 2 1 ⎟ −1 1 0 − 9 − 9 9 ⎠; A = ⎝ − 9 0 ! 2 9 − 49 − 19 − 59 − 19 ⎞ ⎛ 1 0 0 1 row ⎟ ⎜ 0 ⎠ −−−−−−→ ⎝ 0 1 0 operations 0 0 0 1 0 0 1 3 ⎟ ⎠ 2 3 1 −3 − 23 1 3 2 −3 0 ⎞ ⎟ ⎠; ; * ; 8.6 ⎛ 0 ⎜ A−1 = ⎝ 0 1 3 2 3 1 −3 − 23 1 3 − 32 Inverse of a Matrix ⎞ ⎟ ⎠ 0 ⎞ ⎞ ⎛ 1 0 0 0 1 2 −1 2 row ⎟ ⎟ ⎜ 1 0 ⎠ −−−−−−→ ⎝ 0 1 − 13 − 16 0 ⎠; A is singular. 3 operations 0 0 0 −2 −1 1 0 0 1 ⎞ ⎞ ⎛ ⎛ 1 0 0 0 0 5 6 1 −3 0 −1 1 0 0 row row ⎟ ⎟ ⎜ ⎜ 0 1 0 ⎠ −−−−−−→ ⎝ 0 1 1 0 ⎠ −−−−−−→ ⎝ 0 1 0 2 2 1 1 operations operations 0 0 1 −1 −1 1 0 0 1 −1 −1 0 0 1 ⎞ 6 −3 ⎟ 2 −1 ⎠ −1 1 ⎛ ⎞ ⎞ ⎞ ⎛ 5 5 0 0 1 −2 1 0 0 1 −2 8 8 row ⎜ ⎟ ⎟ ⎟ ⎜ 1 0 ⎠ −−−−−−→ ⎝ 0 1 0 0 1 − 21 ⎠; A−1 = ⎝ 0 1 − 21 ⎠ operations 1 1 0 0 1 0 0 0 0 0 0 8 0 0 1 8 8 ⎛ ⎞ ⎞ ⎛ 1 0 0 0 1 2 3 1 1 0 0 0 1 2 3 1 ⎜ −1 0 ⎟ ⎜ 5 1 1 2 1 0 1 0 0⎟ 1 0 0⎟ row ⎜ ⎟ ⎜0 1 2 2 2 25. ⎜ ⎟ ⎟ −−−−−−→ ⎜ 1 2 2 ⎝ 2 1 −3 0 0 0 1 0 ⎠ operations ⎝ 0 0 1 − 3 ⎠ −1 − 0 3 3 1 1 1 0 0 0 1 −2 1 1 2 1 0 0 0 1 1 2 2 ⎛ ⎞ ⎛ 1 2 1 2 1 7 7 ⎞ − 2 − 3 − 16 1 0 0 0 −2 −3 −6 6 6 ⎜ ⎜ 1 4 ⎟ 4 ⎟ 1 1 1 ⎜ ⎟ ⎜ 0 1 0 0 1 − − 1 row 3 3 3 ⎟ 3 3 3 ⎟ ⎟ ; A−1 = ⎜ −−−−−−→ ⎜ ⎜ ⎜ 1 1 1 ⎟ 1 ⎟ 1 1 operations ⎝ 0 0 1 0 0 −3 −3 ⎝ 0 −3 −3 3 ⎠ 3 ⎠ ⎛ 2 4 −2 ⎜ 22. ⎝ 4 2 −2 8 10 −6 ⎛ −1 3 0 ⎜ 23. ⎝ 3 −2 1 0 1 2 ⎛ 5 ⎜ −1 A =⎝ 2 −1 ⎛ 1 2 3 1 ⎜ 24. ⎝ 0 1 4 0 1 0 0 1 0 ⎛ 1 0 ⎜0 ⎜ 26. ⎜ ⎝0 0 0 ⎞ 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 ) 1 −3 27. (AB)−1 = B−1 A−1 = 28. (AB)−1 = B−1 A−1 29. A = (A−1 )−1 = T 1 1 1 2 2 ⎛ 1 0 0 0 0 ⎜0 1 0 0 0⎟ row ⎜ ⎟ ⎟ −−−−−−→ ⎜ 0 ⎠ interchange ⎝ 0 0 1 0 0 30. A = ! 1 2 4 10 31. Multiplying ! " ! ; −1 ⎛ −1 ⎜ =⎝ 2 −2 3 3 −4 T −1 (A ) 4 −3 x −4 "! 3 " = − 21 1 0 1 3 10 3 0 0 1 − 12 ⎞ 1 0 0 0 0 0 0 0 1 0 0 0 1 1⎟ ⎟ ⎟; A−1 0⎠ * 1 2 1 0 1 2 1 ⎜0 ⎜ =⎜ ⎝0 0 0 0 1 0 ⎛ ⎞ −4 20 ⎟ 6 −30 ⎠ 6 ! −32 5 −2 4 −3 x −4 " = −1 1 2 ! " ; A −1 = ! 5 −1 16 − 3x 0 0 16 − 3x " 391 −2 1 2 " ; (A −1 T ) = we see that x = 5. ! 5 −2 −1 1 2 " 0 0 0 1 ⎞ 0 1⎟ ⎟ ⎟ 0⎠ 0 −3 ⎞ ⎟ −1 ⎠; 1 8.6 Inverse of a Matrix 32. A−1 = ! sin θ cos θ 33. (a) AT = ! − cos θ sin θ sin θ cos θ " − cos θ sin θ " ⎛ = A−1 ⎜ (b) AT = ⎝ √1 3 0 − √26 √1 3 √1 2 √1 6 √1 3 − √12 √1 6 ⎞ ⎟ ⎠ = A−1 34. Since det A · det A−1 = det AA−1 = det I = 1, we see that det A−1 = 1/ det A. If A is orthogonal, det A = det AT = det A−1 = 1/ det A and (det A)2 = 1, so det A = ±1. 35. Since A and B are nonsingular, det AB = det A · det B ̸= 0, and AB is nonsingular. 36. Suppose A is singular. Then det A = 0, det AB = det A · det B = 0, and AB is singular. 37. Since det A · det A−1 = det AA−1 = det I = 1, det A−1 = 1/ det A. 38. Suppose A2 = A and A is nonsingular. Then A2 A−1 = AA−1 , and A = I. Thus, if A2 = A, either A is singular or A = I. 39. If A is nonsingular, then A−1 exists, and AB = 0 implies A−1 AB = A−1 0, so B = 0. 40. If A is nonsingular, A−1 exists, and AB = AC implies A−1 AB = A−1 AC, so B = C. ! " ! " 1 0 0 0 41. No, consider A = and B = . 0 0 0 1 42. A is nonsingular if a11 a22 a33 = 0 or a11 , a22 , and a33 are all nonzero. ⎞ ⎛ 1/a11 0 0 ⎟ ⎜ 1/a22 A−1 = ⎝ 0 0 ⎠ 0 0 1/a33 For any diagonal matrix, the inverse matrix is obtaining by taking the reciprocals of the diagonal entries and 43. 44. 45. 46. 47. 48. leaving all other entries 0. )1 * " ! " ! 1 6 4 3 3 −1 A−1 = 2 ; A ; x1 = 6, x2 = −2 = −2 14 − 31 3 ) 2 1* " ! 1" ! 2 3 1 3 6 2 −1 −1 ; A = ; x1 = , x2 = − A = 2 2 −5 − 32 − 13 61 ) 1 3* ! " ) 3* 6 1 3 16 8 4 −1 −1 A = ; A ; x1 = , x2 = − = 1 1 1 4 2 1 −8 4 −2 ! " " ! " ! 4 −11 −2 1 15 −1 A−1 = ; A ; x1 = −11, x2 = = 1 15 3 2 −2 −3 2 2 ⎛ 1 ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 −4 −5 2 5 5 ⎜ ⎟ ⎟ ⎟ ⎜ −1 −1 ⎜ 1 0 ⎠; A ⎝ 0 ⎠ = ⎝ 4 ⎠; x1 = 2, x2 = 4, x3 = −6 A = ⎝ −1 6 − 51 − 15 −6 6 5 ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ 5 1 1 − 12 − 12 1 12 4 3 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ 1 0 ⎠; A−1 ⎝ 2 ⎠ = ⎝ 0 ⎠; x1 = − , x2 = 0, x3 = A−1 = ⎝ − 23 3 2 2 3 5 −3 −1 −1 2 12 12 4 392 8.6 ⎛ ⎜ 49. A−1 = ⎝ 50. A−1 −2 1 4 5 4 −3 − 41 7 4 2 −1 ⎜ −1 2 ⎜ =⎜ ⎝ 1 −1 ⎛ Inverse of a Matrix ⎞ ⎛ ⎞ ⎞ ⎛ 2 1 21 ⎜ ⎟ ⎟ ⎜ 0⎟ ⎠; A−1 ⎝ −3 ⎠ = ⎝ 1 ⎠; x1 = 21, x2 = 1, x3 = −11 −11 7 −1 ⎞ ⎞ ⎞ ⎛ ⎛ 1 2 1 1 ⎜ 1⎟ ⎜ 2⎟ −1 −1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟; x1 = 1, x2 = 2, x3 = −1, x4 = −4 ⎟=⎜ ⎟; A−1 ⎜ ⎝ −5 ⎠ ⎝ −1 ⎠ 1 1⎠ −4 3 1 −1 1 0 ) 1 * "! " ! " ! " ) 9 * ! " ! " 1 x1 7 −2 b1 5 10 6 10 10 10 −1 −1 −1 51. ; A = ; X=A = ; = 13 ; X = A = 3 7 3 2 b2 x2 16 4 50 − 20 20 20 " " ! ! −2 0 −1 X=A = −7 −20 ⎞ ⎛ ⎛ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ⎞ 1 2 5 b1 −1 x1 2 −1 −1 −12 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ 52. ⎝ 2 3 8 ⎠ ⎝ x2 ⎠ = ⎝ b2 ⎠; A−1 = ⎝ 12 −7 −2 ⎠; X = A−1 ⎝ 4 ⎠ = ⎝ −52 ⎠; −1 1 2 −5 3 1 6 b3 x3 23 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ⎞ ⎛ 0 1 0 3 ⎜ ⎟ ⎟ ⎟ ⎜ −1 ⎜ −1 ⎜ ⎟ X = A ⎝ 3 ⎠ = ⎝ 9 ⎠; = A ⎝ −5 ⎠ = ⎝ 27 ⎠ ! 3 −3 −11 4 53. det A = 18 ̸= 0, so the system has only the trivial solution. 54. det A = 0, so the system has a nontrivial solution. 55. det A = 0, so the system has a nontrivial solution. 56. det A = 12 ̸= 0, so the system has only the trivial solution. ⎛ ⎞ ⎞⎛ ⎞ ⎛ 1 1 1 0 i1 ⎜ ⎟ ⎟⎜ ⎟ ⎜ 57. (a) ⎝ −R1 R2 0 ⎠ ⎝ i2 ⎠ = ⎝ E2 − E1 ⎠ E3 − E2 i3 0 −R2 R3 (b) det A = R1 R2 + R1 R3 + R2 R3 > 0, so A is nonsingular. ⎞ ⎛ R2 R3 −R2 − R3 −R2 1 ⎟ ⎜ R3 −R1 ⎠; (c) A−1 = ⎝ R1 R3 R1 R2 + R1 R3 + R2 R3 R1 R2 R2 R1 + R 2 ⎞ ⎛ ⎞ ⎛ R 2 E1 − R 2 E 3 + R 3 E 1 − R 3 E 2 0 1 ⎟ ⎜ ⎟ ⎜ A−1 ⎝ E2 − E1 ⎠ = ⎝ R 1 E2 − R 1 E 3 − R 3 E 1 + R 3 E 2 ⎠ R1 R2 + R1 R3 + R2 R3 −R1 E2 + R1 E3 − R2 E1 + R2 E3 E3 − E2 58. (a) We write the equations in the form −4u1 + u2 + u4 = −200 u1 − 4u2 + u3 = −300 u2 − 4u3 + u4 = −300 u1 + u3 − 4u4 = −200. ⎞ ⎞⎛ ⎞ ⎛ −200 −4 1 0 1 u1 ⎜ 1 −4 ⎟ ⎜ ⎟ ⎜ 1 0⎟ ⎜ ⎟ ⎜ u2 ⎟ ⎜ −300 ⎟ In matrix form this becomes ⎜ ⎟. ⎟⎜ ⎟ = ⎜ ⎝ 0 1 −4 1 ⎠ ⎝ u3 ⎠ ⎝ −300 ⎠ −200 1 0 1 −4 u4 ⎛ 393 8.6 Inverse of a Matrix (b) A −1 ⎛ 7 − 24 ⎜ 1 ⎜ − 12 =⎜ ⎜ 1 ⎝ − 24 1 − 12 1 − 12 1 − 24 1 − 12 7 − 24 7 − 24 1 − 24 1 − 12 1 − 12 1 − 12 ⎞ ⎛ 225 ⎞ −200 2 ⎜ 275 ⎟ 1 ⎟ ⎟ ⎜ − 24 ⎟ −300 ⎟ ⎜ 2 ⎟ ⎟; A−1 ⎜ ⎟; u1 = u4 = 225 , u2 = u3 = 275 ⎟=⎜ ⎜ ⎜ ⎟ 275 1 ⎟ 2 2 ⎠ ⎝ −300 − 12 ⎠ ⎝ 2 ⎠ 225 −200 −7 ⎞ ⎛ 24 2 EXERCISES 8.7 Cramer’s Rule 1. det A = 10, det A1 = −6, det A2 = 12; x1 = −6 10 2. det A = −3, det A1 = −6, det A2 = −6; x1 = = − 35 , x2 = −6 −3 −6 −3 = 2, x2 = 3. det A = 0.3, det A1 = 0.03, det A2 = −0.09; x1 = 0.03 0.3 12 10 −0.09 0.3 −0.00315 −0.015 = −0.3 = 0.21, x2 = , s= 35 −70 = − 21 7. det A = 11, det A1 = −44, det A2 = 44, det A3 = −55; x1 = −44 11 = −4, x2 = 6. det A = −70, det A1 = −14, det A2 = 35; r = −14 −70 = 1 5 6 5 =2 = 0.1 , x2 = 4. det A = −0.015, det A1 = −0.00315, det A2 = −0.00855; x1 = 5. det A = 1, det A1 = 4, det A2 = −7; x = 4, y = −7 = 173 8. det A = −63, det A1 = 173, det A2 = −136, det A3 = − 61 2 ; x1 = − 63 , x2 = 9. det A = −12, det A1 = −48, det A2 = −18, det A3 = −12; u = 48 12 = 4, v = 44 11 18 12 −0.00855 −0.015 = 4, x3 = 136 63 = , x3 = 3 2 = 0.57 −55 11 61 126 = −5 , w=1 10. det A = 1, det A1 = −2, det A2 = 2, det A3 = 5; x = −2, y = 2, z = 5 11. det A = 6 − 5k, det A1 = 12 − 7k, det A2 = 6 − 7k; x1 = for k = 6/5. 12 − 7k 6 − 7k , x2 = . The system is inconsistent 6 − 5k 6 − 5k ϵ−1−1 1 1 ϵ−2 = =1− , x2 = ϵ−1 ϵ−1 ϵ−1 ϵ−1 (b) When ϵ = 1.01, x1 = −99 and x2 = 100. When ϵ = 0.99, x1 = 101 and x2 = −100. 12. (a) det A = ϵ − 1, det A1 = ϵ − 2, det A2 = 1; x1 = 13. det A ≈ 0.6428, det A1 ≈ 289.8, det A2 ≈ 271.9; x1 ≈ 289.8 0.6428 ≈ 450.8, x2 ≈ 271.9 0.6428 ≈ 423 14. We have (sin 30◦ )F + (sin 30◦ )(0.5N ) + N sin 60◦ = 400 and (cos 30◦ )F + (cos 30◦ )(0.5N ) − N cos 60◦ = 0. The system is (sin 30◦ )F + (0.5 sin 30◦ + sin 60◦ )N = 400 (cos 30◦ )F + (0.5 cos 30◦ − cos 60◦ )N = 0. det A ≈ −1, det A1 ≈ −26.795, det A2 ≈ −346.41; F ≈ 26.795, N ≈ 346.41 15. The system is i1 + i2 − i3 = 0 r1 i1 − r2 i2 = E1 − E2 r2 i2 + Ri3 = E2 det A = −r1 R − r2 R − r1 r2 , det A3 = −r1 E2 , −r2 E1 ; i3 = 394 r1 E2 + r2 E1 r1 R + r2 R + r1 r2 8.8 The Eigenvalue Problem EXERCISES 8.8 The Eigenvalue Problem 1. K3 since 2. K1 3. K3 4. K2 5. K2 ! 4 5 2 1 "! −2 5 " = ! 2 −5 " " −2 ; 5 √ ! 2 = (−1) ! λ = −1 ! "! " " " √ √ 1 1 2 −1 √ √ , λ= 2 √ and K2 since = = 2 2− 2 −2 + 2 2 2− 2 2 −2 √ " ! √ " √ " "! ! ! √ √ 2+ 2 2 −1 2+2 2 2+ 2 √ = 2 = ; λ= 2 2 −2 2 2 2 2 ! " ! " "! " ! 6 3 0 −5 −5 since = ; λ=0 =0 10 2 1 10 0 " " ! " ! "! ! 2 + 2i −4 + 4i 2 + 2i 2 8 ; λ = 2i = 2i = since −1 −2i −1 −1 −2 ⎞ ⎛ ⎞ ⎞ ⎛ ⎞⎛ ⎛ 4 12 4 1 −2 2 ⎟ ⎜ ⎟ ⎟ ⎜ ⎟⎜ ⎜ 1 −2 ⎠ ⎝ −4 ⎠ = ⎝ −12 ⎠ = 3 ⎝ −4 ⎠; λ = 3 and K3 since ⎝ −2 ! 0 0 ⎞ −1 1 −2 2 −1 ⎟⎜ ⎜ ⎟ ⎜ ⎟ 1 −2 ⎠ ⎝ 1 ⎠ = ⎝ 1 ⎠; λ = 1 ⎝ −2 2 2 1 1 1 ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎞ 1 3 1 1 0 ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ 2 1 ⎠ ⎝ 4 ⎠ = ⎝ 12 ⎠ = 3 ⎝ 4 ⎠; λ = 3 2 2 ⎛ −1 ⎜ 6. K2 since ⎝ 1 0 0 1 ⎞⎛ ⎛ ⎞ ⎛ 3 - −1 − λ 2 - = (λ − 6)(λ − 1) = 0. 7. We solve det(A − λI) = −7 8 − λ" ! " ! 1 −2/7 0 −7 2 0 For λ1 = 6 we have =⇒ 0 −7 2 0 0 0 ! " 2 so that k1 = 72 k2 . If k2 = 7 then K1 = . For λ2 = 1 we have 7 ! " ! " −2 2 0 1 −1 0 =⇒ −7 7 0 0 0 0 ! " 1 so that k1 = k2 . If k2 = 1 then K2 = . 1 -2 − λ 1 -= λ(λ − 3) = 0. 8. We solve det(A − λI) = 2 1 − λ! " ! " 2 1 0 1 1/2 0 For λ1 = 0 we have =⇒ 2 1 0 0 0 0 3 −1 3 9 395 8.8 The Eigenvalue Problem " −1 . For λ2 = 3 we have 2 " ! " ! 1 −1 0 −1 1 0 =⇒ 2 −2 0 0 0 0 ! " 1 so that k1 = k2 . If k2 = 1 then K2 = . 1 - −8 − λ −1 - = (λ + 4)2 = 0. We solve det(A − λI) = -16 −λ ! " ! " −4 −1 0 1 1/4 0 For λ1 = λ2 = −4 we have =⇒ 16 4 0 0 0 0 ! " −1 so that k1 = − 41 k2 . If k2 = 4 then K1 = . 4 -1 − λ 1 -= (λ − 3/2)(λ − 1/2) = 0. We solve det(A − λI) = -1/4 1 − λ ! " ! " −1/2 1 1 −2 0 0 For λ1 = 3/2 we have =⇒ 1/4 −1/2 0 0 0 0 ! " 2 so that k1 = 2k2 . If k2 = 1 then K1 = . If λ2 = 1/2 then 1 " ! " ! 1/2 1 0 1 2 0 =⇒ 1/4 1/2 0 0 0 0 ! " −2 so that k1 = −2k2 . If k2 = 1 then K2 = . 1 - −1 − λ 2 -= λ2 + 9 = (λ − 3i)(λ + 3i) = 0. We solve det(A − λI) = -−5 1 − λ" ! " ! 1 −(1/5) + (3/5)i 0 0 −1 − 3i 2 =⇒ For λ1 = 3i we have 0 −5 1 − 3i 0 0 0 " ! + , 1 − 3i so that k1 = 51 − 53 i k2 . If k2 = 5 then K1 = . For λ2 = −3i we have 5 " ! " ! 0 1 − 15 − 53 i 0 −1 + 3i 2 =⇒ −5 1 + 3i 0 0 0 0 " ! , + 1 + 3i . so that k1 = 15 + 53 i k2 . If k2 = 5 then K2 = 5 -1 − λ −1 -= λ2 − 2λ + 2 = 0. We solve det(A − λI) = 1 1 − λ! " ! " i −1 0 i −1 0 For λ1 = 1 − i we have =⇒ 1 i 0 0 0 0 ! " ! " i −i so that k1 = −ik2 . If k2 = 1 then K1 = . and K2 = K1 = 1 1 so that k1 = − 21 k2 . If k2 = 2 then K1 = 9. 10. 11. 12. ! 396 8.8 The Eigenvalue Problem -4 − λ 8 - = (λ − 4)(λ + 5) = 0. det(A − λI) = -0 −5 − λ " ! " ! 0 1 0 0 8 0 =⇒ For λ1 = 4 we have 0 −9 0 0 0 0 ! " 1 so that k2 = 0. If k1 = 1 then K1 = . For λ2 = −5 we have 0 " ! " ! 1 98 0 9 8 0 =⇒ 0 0 0 0 0 0 ! " −8 so that k1 = − 98 k2 . If k2 = 9 then K2 = . 9 -7 − λ 0 -= (λ − 7)(λ − 13) = 0. 14. We solve det(A − λI) = 0 13 − λ ! " ! " 0 0 0 0 1 0 For λ1 = 7 we have =⇒ 0 6 0 0 0 0 ! " 1 so that k2 = 0. If k1 = 1 then K1 = . For λ2 = 13 we have 0 ! " ! " 1 0 0 −6 0 0 =⇒ 0 0 0 0 0 0 ! " 0 so that k1 = 0. If k2 = 1 then K2 = . 1 - -5 − λ −1 0 -−1 0 -- -- 4 − λ - 9 - = λ(4 − λ)(λ + 4) = 0. −5 − λ 9 - = - 4 − λ −5 − λ 15. We solve det(A − λI) = - 0 - - 5 −1 −λ −1 −λ - - 4 − λ ⎛ ⎛ ⎞ ⎞ 5 −1 0 0 1 0 −9/25 0 ⎜ ⎜ ⎟ ⎟ For λ1 = 0 we have ⎝ 0 −5 9 0 ⎠ =⇒ ⎝ 0 1 −9/5 0 ⎠ 13. We solve 5 −1 0 0 ⎛ 0 0 ⎞ 0 0 9 ⎜ ⎟ and k2 = If k3 = 25 then K1 = ⎝ 45 ⎠. If λ2 = 4 then so that k1 = 25 ⎛ ⎛ ⎞ ⎞ 1 −1 0 0 1 0 −1 0 ⎜ ⎜ ⎟ ⎟ 9 0 ⎠ =⇒ ⎝ 0 1 −1 0 ⎠ ⎝ 0 −9 5 −1 −4 0 0 0 0 0 ⎛ ⎞ 1 ⎜ ⎟ so that k1 = k3 and k2 = k3 . If k3 = 1 then K2 = ⎝ 1 ⎠. If λ3 = −4 then 1 ⎛ ⎛ ⎞ ⎞ 9 −1 0 0 1 0 −1 0 ⎜ ⎜ ⎟ ⎟ ⎝ 0 −1 9 0 ⎠ =⇒ ⎝ 0 1 −9 0 ⎠ 9 25 k3 9 5 k3 . 5 −1 4 0 0 ⎛ ⎞ 1 ⎜ ⎟ so that k1 = k3 and k2 = 9k3 . If k3 = 1 then K3 = ⎝ 9 ⎠. 1 397 0 0 0 8.8 The Eigenvalue Problem -3 − λ 0 0 -16. We solve det(A − λI) = - 0 2−λ 0 - = (3 − λ)(2 − λ)(1 − λ) = 0. - 4 0 1 − λ⎛ ⎞ ⎞ ⎛ 1 0 0 0 2 0 0 0 ⎜ ⎟ ⎟ ⎜ For λ1 = 1 we have ⎝ 0 1 0 0 ⎠ =⇒ ⎝ 0 1 0 0 ⎠ 4 0 0 0 0 0 0 0 ⎛ ⎞ 0 ⎜ ⎟ so that k1 = 0 and k2 = 0. If k3 = 1 then K1 = ⎝ 0 ⎠. If λ2 = 2 then 1 ⎛ ⎛ ⎞ ⎞ 1 0 0 0 1 0 0 0 ⎜ ⎜ ⎟ ⎟ 0 0 ⎠ =⇒ ⎝ 0 0 1 0 ⎠ ⎝0 0 4 0 −1 0 0 0 0 0 ⎛ ⎞ 0 ⎜ ⎟ so that k1 = 0 and k3 = 0. If k2 = 1 then K2 = ⎝ 1 ⎠. If λ3 = 3 then 0 ⎛ ⎞ ⎞ ⎛ 1 0 −1/2 0 0 0 0 0 ⎜ ⎟ ⎟ ⎜ 0 0 ⎠ =⇒ ⎝ 0 1 0 0⎠ ⎝ 0 −1 4 0 −2 0 0 0 0 0 ⎛ ⎞ 1 ⎜ ⎟ 1 so that k1 = 2 k3 and k2 = 0. If k3 = 2 then K3 = ⎝ 0 ⎠. 2 - −λ 4 0 -17. We solve det(A − λI) = - −1 −4 − λ 0 - = −(λ + 2)3 = 0. - 0 0 −2 − λ ⎞ ⎞ ⎛ ⎛ 1 2 0 0 2 4 0 0 ⎟ ⎟ ⎜ ⎜ For λ1 = λ2 = λ3 = −2 we have ⎝ −1 −2 0 0 ⎠ =⇒ ⎝ 0 0 0 0 ⎠ 0 0 0 0 0 0 0 0 so that k1 = −2k2 . If k2 = 1 and k3 = 1 then ⎛ ⎞ −2 ⎟ ⎜ K1 = ⎝ 1 ⎠ 0 18. We solve -1 − λ det(A − λI) = - 0 - 0 For λ1 = 3 we have 6 ⎛ ⎞ 0 ⎜ ⎟ and K2 = ⎝ 0 ⎠ . 1 - - -1 − λ - - 1 -=- 0 - 2 − λ- - 0 0 2−λ 1 ⎛ −2 6 0 ⎜ 0 ⎝ 0 0 0 1 −1 3 − λ 3 − λ - = (3 − λ)(1 − λ)2 = 0. 1 2 − λ⎛ ⎞ ⎞ 0 1 0 −3 0 ⎜ ⎟ ⎟ 0 ⎠ =⇒ ⎝ 0 1 −1 0 ⎠ 0 0 0 0 0 398 6 0 8.8 The Eigenvalue Problem ⎛ ⎞ 3 ⎜ ⎟ so that k1 = 3k3 and k2 = k3 . If k3 = 1 then K1 = ⎝ 1 ⎠. For λ2 = λ3 = 1 we have 1 ⎛ ⎛ ⎞ ⎞ 0 6 0 0 0 1 0 0 ⎜ ⎜ ⎟ ⎟ ⎝ 0 1 1 0 ⎠ =⇒ ⎝ 0 0 1 0 ⎠ 0 1 1 0 0 0 0 0 ⎛ ⎞ 1 ⎜ ⎟ so that k2 = 0 and k3 = 0. If k1 = 1 then K2 = ⎝ 0 ⎠. 0 - −λ 0 −1 -19. We solve det(A − λI) = - 1 −λ 0 - = −(λ + 1)(λ2 + 1) = 0. - 1 1 −1 − λ ⎛ ⎞ ⎞ ⎛ 1 0 −1 0 1 0 −1 0 ⎜ ⎟ ⎟ ⎜ 0 0 ⎠ =⇒ ⎝ 0 1 1 0⎠ For λ1 = −1 we have ⎝1 1 1 1 0 0 0 0 0 0 ⎞ ⎛ 1 ⎟ ⎜ so that k1 = k3 and k2 = −k3 . If k3 = 1 then K1 = ⎝ −1 ⎠. For λ2 = i we have 1 ⎛ −i 0 −1 ⎜ 0 ⎝ 1 −i 1 1 −1 − i 0 ⎛ ⎞ 1 0 −i 0 ⎞ ⎜ ⎟ ⎟ 0 ⎠ =⇒ ⎝ 0 1 −1 0 ⎠ 0 0 0 0 0 ⎛ ⎞ ⎛ ⎞ i −i ⎜ ⎟ ⎜ ⎟ so that k1 = ik3 and k2 = k3 . If k3 = 1 then K2 = ⎝ 1 ⎠ and K3 = K2 = ⎝ 1 ⎠. 1 1 20. We solve -2 − λ det(A − λI) = - 5 - 0 For λ1 = 2 we have −1 0 -2−λ 4 - = −λ3 + 6λ2 − 13λ + 10 = (λ − 2)(−λ2 + 4λ − 5) 1 2 − λ= (λ − 2)(λ − (2 + i))(λ − (2 − i)) = 0. ⎛ 0 −1 0 ⎜ 0 4 ⎝5 0 1 0 0 ⎞ ⎛ 1 0 ⎜ ⎟ 0 ⎠ =⇒ ⎝ 0 1 0 0 0 ⎛ −4 ⎞ 4/5 0 0 0 ⎞ ⎟ 0⎠ 0 ⎜ ⎟ so that k1 = − 45 k3 and k2 = 0. If k3 = 5 then K1 = ⎝ 0 ⎠. For λ2 = 2 + i we have 5 ⎛ ⎞ ⎞ ⎛ 1 −i 0 0 −i −1 0 0 ⎜ ⎟ ⎟ ⎜ 4 0 ⎠ =⇒ ⎝ 0 1 −i 0 ⎠ ⎝ 5 −i 0 1 −i 0 0 0 0 0 399 8.8 The Eigenvalue Problem ⎛ −i ⎞ ⎟ ⎜ so that k1 = ik2 and k2 = ik3 . If k3 = i then K2 = ⎝ −1 ⎠. For λ3 = 2 − i we have i ⎛ ⎛ ⎞ ⎞ i −1 0 0 1 i 0 0 ⎜ ⎜ ⎟ ⎟ i 4 0 ⎠ =⇒ ⎝ 0 1 i 0 ⎠ ⎝5 0 1 i 0 ⎛ 0 0 ⎞ 0 0 −1 ⎟ ⎜ so that k1 = −ik2 and k2 = −ik3 . If k3 = i then K3 = ⎝ 1 ⎠. i -1 − λ 2 3 -21. We solve det(A − λI) = - 0 5−λ 6 - = −(λ − 1)(λ − 5)(λ + 7) = 0. - 0 0 −7 − λ ⎛ ⎞ ⎞ ⎛ 0 1 0 0 0 2 3 0 ⎜ ⎟ ⎟ ⎜ 6 0 ⎠ =⇒ ⎝ 0 0 1 0 ⎠ For λ1 = 1 we have ⎝0 4 0 0 −6 0 0 0 0 0 ⎛ ⎞ 1 ⎜ ⎟ so that k2 = k3 = 0. If k1 = 1 then K1 = ⎝ 0 ⎠. For λ2 = 5 we have 0 ⎛ ⎛ ⎞ ⎞ −4 2 3 0 1 − 21 0 0 ⎜ ⎜ ⎟ ⎟ 6 0 ⎠ =⇒ ⎝ 0 0 1 0⎠ ⎝ 0 0 0 0 −12 0 0 0 0 0 ⎛ ⎞ 1 ⎜ ⎟ so that k3 = 0 and k2 = 2k1 . If k1 = 1 then K2 = ⎝ 2 ⎠. For λ3 = −7 we have 0 ⎛ ⎛ ⎞ ⎞ 8 2 3 0 1 0 14 0 ⎜ ⎜ ⎟ ⎟ ⎝ 0 12 6 0 ⎠ =⇒ ⎝ 0 1 12 0 ⎠ 0 0 0 0 0 0 0 0 ⎞ ⎛ −1 ⎟ ⎜ so that k1 = − 41 k3 and k2 = − 12 k3 . If k3 = 4 then K3 = ⎝ −2 ⎠. 22. We solve For λ1 = λ2 = 0 we have 4 0 0 -−λ 0 - = −λ2 (λ − 1) = 0. 0 1 − λ⎛ ⎞ ⎞ 0 0 1 0 0 0 0 ⎜ ⎟ ⎟ 0 0 0 ⎠ =⇒ ⎝ 0 0 0 0 ⎠ 0 1 0 0 0 0 0 ⎛ ⎞ ⎛ ⎞ 1 0 ⎜ ⎟ ⎜ ⎟ = ⎝ 0 ⎠ and if k1 = 0 and k2 = 1 then K2 = ⎝ 1 ⎠. For λ3 = 1 0 0 - −λ det(A − λI) = - 0 - 0 ⎛ 0 ⎜ 0 ⎝ 0 so that k3 = 0. If k1 = 1 and k2 = 0 then K1 400 8.8 we have ⎛ −1 ⎜ ⎝ 0 0 0 0 −1 0 0 ⎛ ⎞ 1 0 0 ⎜ ⎟ 0 ⎠ =⇒ ⎝ 0 1 0 0 0 0 ⎛ ⎞ 0 ⎜ ⎟ so that k1 = k2 = 0. If k3 = 1 then K3 = ⎝ 0 ⎠. 1 ! " 5 1 23. The eigenvalues and eigenvectors of A = are 1 5 0 0 0 0 The Eigenvalue Problem ⎞ ⎟ 0⎠ 0 " ! " 1 1 , K2 = −1 1 ! " 5 −1 1 are and the eigenvalues and eigenvectors of A−1 = 24 −1 5 " ! " ! 1 1 1 1 λ 1 = , λ2 = , K 1 = . , K2 = 4 6 1 −1 ⎛ ⎞ 1 2 −1 ⎜ ⎟ 0 1 ⎠ are 24. The eigenvalues and eigenvectors of A = ⎝ 1 4 −4 5 ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ −1 −2 −1 ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ λ1 = 1, λ2 = 2, λ3 = 3, K1 = ⎝ 1 ⎠ , K2 = ⎝ 1 ⎠ , K3 = ⎝ 1 ⎠ . 2 4 4 ⎞ ⎛ 4 −6 2 1⎜ ⎟ −1 9 −2 ⎠ are and the eigenvalues and eigenvectors of A = ⎝ −1 6 −4 12 −2 ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ −1 −2 −1 1 1 ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ λ1 = 1, λ2 = , λ3 = , K1 = ⎝ 1 ⎠ , K2 = ⎝ 1 ⎠ , K3 = ⎝ 1 ⎠ . 2 3 2 4 4 λ1 = 4, -6 25. Since det A = -3 λ2 = 6, K1 = ! 0 -= 0 the matrix is singular. Now from 0-6 − λ 0 - = λ(λ − 6) det(A − λI) = -3 −λ - we see λ = 0 is an eigenvalue. -1 0 1 -26. Since det A = - 4 −4 5 - = 0 the matrix is singular. Now from - 7 −4 8 -1 − λ 0 1 -det(A − λI) = - 4 −4 − λ 5 - = −λ(λ2 − 5λ − 15) - 7 −4 8 − λwe see λ = 0 is an eigenvalue. 401 8.8 The Eigenvalue Problem 27. (a) Since p + 1 − p = 1 and q + 1 − q = 1, the first matrix A is stochastic. Since and 1 6 + 1 3 + 1 2 1 2 + 1 4 + 1 4 = 1, 1 3 + 1 3 + 1 3 = 1, = 1, the second matrix A is stochastic. (b) The matrix from part (a) is shown with its eigenvalues and corresponding eigenvectors. ⎛1 1 1⎞ 2 4 4 √ 1 √ ⎜1 1 1⎟ 1 1 2 , 6 + 12 2; ⎝ 3 3 3 ⎠; eigenvalues: 1, 61 − 12 1 6 1 3 1 2 √ √ + , + 3(1+√2) 2(−2+√2) , 2) 2(2+ 2) √ , √ ,1 , − √ , √ eigenvectors: (1, 1, 1), − 3(−1+ ,1 −6+ 2 −6+ 2 6+ 2 6+ 2 Further examples indicate that 1 is always an eigenvalue with corresponding eigenvector (1, 1, 1). To prove this, let A be a stochastic matrix and K = (1, 1, 1). Then ⎞ ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎛ a11 + · · · + a1n a11 · · · a1n 1 1 .. .. ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ .. AK = ⎝ . ⎠ = ⎝ . ⎠ = 1K, . . ⎠⎝ . ⎠ = ⎝ 1 1 an1 + · · · + ann an1 · · · ann and 1 is an eigenvalue of A with corresponding eigenvector (1, 1, 1). (c) For the 3 × 3 matrix in part (a) we have ⎛ 3 7 ⎜ A2 = ⎝ 8 1 3 5 18 24 11 36 23 72 1 3 13 36 29 72 ⎞ ⎟ ⎠, ⎛ ⎜ A3 = ⎝ 49 144 71 216 5 16 29 96 11 36 67 216 103 288 79 216 163 432 ⎞ ⎟ ⎠. These powers of A are also stochastic matrices. To prove that this is true in general for 2 × 2 matrices, we prove the more general theorem that any product of 2 × 2 stochastic matrices is stochastic. Let ! " " ! a11 a12 b11 b12 A= and B = b21 b22 a21 a22 be stochastic matrices. Then AB = ! a11 b11 + a12 b21 a11 b12 + a12 b22 a21 b11 + a22 b21 a21 b12 + a22 b22 " . The sums of the rows are a11 b11 + a12 b21 + a11 b12 + a12 b22 = a11 (b11 + b12 ) + a12 (b21 + b22 ) = a11 (1) + a12 (1) = a11 + a12 = 1 a21 b11 + a22 b21 + a21 b12 + a22 b22 = a21 (b11 + b12 ) + a22 (b21 + b22 ) = a21 (1) + a22 (1) = a21 + a22 = 1. Thus, the product matrix AB is stochastic. It follows that any power of a 2 × 2 matrix is stochastic. The proof in the case of an n × n matrix is very similar. 402 8.9 Powers of Matrices EXERCISES 8.9 Powers of Matrices 1. The characteristic equation is λ2 − 6λ + 13 = 0. Then " ! ! 6 −7 −12 − A2 − 6A + 13I = 24 24 17 −12 30 " + 2. The characteristic equation is −λ3 + λ2 + 4λ − 1. Then ⎛ ⎞ ⎛ ⎛ ⎞ 2 6 13 1 2 5 0 ⎜ ⎟ ⎜ ⎜ ⎟ 3 2 −A + A + A − I = − ⎝ 4 5 17 ⎠ + ⎝ 0 4 5 ⎠ + 4 ⎝ 1 1 5 9 1 1 4 0 ! 13 0 0 13 " ⎞ ⎛ = ! 0 0 0 0 " ⎞ . ⎛ ⎞ 1 2 1 0 0 0 0 0 0 ⎟ ⎜ 3⎠ + ⎝0 1 ⎟ ⎜ 0⎠ = ⎝0 0 ⎟ 0⎠. 1 1 0 0 1 0 0 0 3. The characteristic equation is λ2 − 3λ − 10 = 0, with eigenvalues −2 and 5. Substituting the eigenvalues into λm = c0 + c1 λ generates (−2)m = c0 − 2c1 5m = c0 + 5c1 . Solving the system gives c0 = 1 [5(−2)m + 2(5)m ], 7 1 [−(−2)m + 5m ]. 7 c1 = Thus A m = c0 I + c1 A = )1 m m+1 7 [3(−1) 2 2 m 7 [−(−2) and 3 A = ! 11 38 3 m m 7 [−(−2) + 5 ] 1 m m 7 [(−2) + 6(5) ] + 5m ] + 5m ] 57 106 " * . 4. The characteristic equation is λ2 − 10λ + 16 = 0, with eigenvalues 2 and 8. Substituting the eigenvalues into λm = c0 + c1 λ generates 2m = c0 + 2c1 8m = c0 + 8c1 . Solving the system gives c0 = 1 m+2 (2 − 8m ), 3 c1 = 1 (−2m + 8m ). 6 Thus A m = c0 I + c1 A = and A4 = ! )1 m 2 (2 1 m 2 (2 2056 −2040 403 + 8m ) − 8m ) −2040 2056 1 m 2 (2 1 m 2 (2 " . − 8m ) + 8m ) * 8.9 Powers of Matrices 5. The characteristic equation is λ2 − 8λ − 20 = 0, with eigenvalues −2 and 10. Substituting the eigenvalues into λm = c0 + c1 λ generates (−2)m = c0 − 2c1 10m = c0 + 10c1 . Solving the system gives c0 = 1 [5(−2)m + 10m ], 6 c1 = 1 [−(−2)m + 10m ]. 12 Thus A m = c0 I + c1 A = 5 m m 12 [−(−2) + 10 ] 1 m m 6 [5(−2) + 10 ] )1 m m m+1 ] 6 [(−2) + 2 5 1 m m 3 [−(−2) + 10 ] * and 5 A = ! 83328 33344 41680 16640 " . 6. The characteristic equation is λ2 + 4λ + 3 = 0, with eigenvalues −3 and −1. Substituting the eigenvalues into λm = c0 + c1 λ generates (−3)m = c0 − 3c1 (−1)m = c0 − c1 . Solving the system gives c0 = 1 [−(−3)m + 3(−1)m ], 2 c1 = 1 [−(−3)m + (−1)m ]. 2 Thus A m = c0 I + c1 A = ) (−1)m −(−3)m + (−1)m (−3)m 0 * and A6 = ! 1 0 −728 729 " . 7. The characteristic equation is −λ3 + 2λ2 + λ − 2 = 0, with eigenvalues −1, 1, and 2. Substituting the eigenvalues into λm = c0 + c1 λ + c2 λ2 generates (−1)m = c0 − c1 + c2 1 = c0 + c1 + c2 2 m = c0 + 2c1 + 4c2 . Solving the system gives 1 [3 + (−1)m − 2m ], 3 1 c1 = [1 − (−1)m ], 2 1 c2 = [−3 + (−1)m + 2m+1 ]. 6 c0 = Thus 404 8.9 ⎛ −1 + 2m 1 ⎜ Am = c0 I + c1 A + c2 A2 = ⎝ 0 1 m m+1 ] 3 [(−1) + 2 1 m m 3 [−(−1) + 2 ] 0 and ⎛ 1 1023 ⎜ A10 = ⎝ 0 0 683 341 1023 −1 + 2m Powers of Matrices ⎞ ⎟ − 32 [(−1)m − 2m ] ⎠ 1 m 3 [2(−1) + 2m ] ⎞ ⎟ 682 ⎠ . 342 √ √ 8. The characteristic equation is −λ3 − λ2 + 2λ + 2 = 0, with eigenvalues −1, − 2 , and 2 . Substituting the eigenvalues into λm = c0 + c1 λ + c2 λ2 generates (−1)m = c0 − c1 + c2 √ √ (− 2 )m = c0 − 2c1 + 2c2 √ √ ( 2 )m = c0 + 2c1 + 2c2 . Solving the system gives √ √ √ √ c0 = [2 − ( 2 )m−1 − ( 2 )m−2 ](−1)m + ( 2 − 1)( 2 )m−2 , √ 1 c1 = [1 − (−1)m ]( 2 )m−1 , 2 √ √ √ 1 √ 1 c2 = (−1)m+1 + (1 + 2 )(−1)m ( 2 )m−1 + ( 2 − 1)( 2 )m−1 . 2 2 Thus Am = c0 I + c1 A + c2 A2 and ⎛ 1 0 ⎜ 6 A = ⎝7 8 0 0 ⎞ 7 ⎟ −7 ⎠ . 8 9. The characteristic equation is −λ3 +3λ2 +6λ−8 = 0, with eigenvalues −2, 1, and 4. Substituting the eigenvalues into λm = c0 + c1 λ + c2 λ2 generates (−2)m = c0 − 2c1 + 4c2 1 = c0 + c1 + c2 4 m = c0 + 4c1 + 16c2 . Solving the system gives 1 [8 + (−1)m 2m+1 − 4m ], 9 1 [4 − 5(−2)m + 4m ], c1 = 18 1 c2 = [−2 + (−2)m + 4m ]. 18 c0 = Thus ⎛1 m 9 [(−2) ⎜ Am = c0 I + c1 A + c2 A2 = ⎝ + (−1)m 2m+1 + 3 · 22m+1 ] − 23 [(−2)m − 4m ] 1 3 [−3 + (−2)m + 22m+1 ] 405 1 m m 3 [−(−2) + 4 ] 1 m m+1 + 4m ] 3 [(−1) 2 1 m m 3 [−(−2) + 4 ] 0 ⎞ ⎟ 0⎠ 1 8.9 Powers of Matrices and A10 ⎛ 699392 ⎜ = ⎝ 698368 699391 ⎞ 0 ⎟ 0⎠. 1 349184 350208 349184 10. The characteristic equation is −λ3 − 23 λ2 + 23 λ + 1 = 0, with eigenvalues −2, − 21 , and 1. Substituting the eigenvalues into λm = c0 + c1 λ + c2 λ2 generates (−2)m = c0 − 2c1 + 4c2 + 1 ,m 1 1 − = c0 − c1 + c2 2 2 4 1 = c0 + c1 + c2 . Solving the system gives 1 −m [2 [(−4)m + 8(−1)m + 2m+1 − (−1)m 22m+1 ], 9 1 c1 = − 2−m [(−4)m + 4(−1)m − 5 · 2m ], 9 2 c2 = [1 + (−2)m − (−1)m 2m−1 ]. 9 c0 = Thus Am = c0 I + c1 A + c2 A2 ⎛ 1 and ⎜ =⎝ −m [2(−1)m + 2m ] 32 . + 1 ,m / 2 3 −1 + − 2 m m m − 19 2−m [7(−4) − 6(−1) − 3 · 2 + (−1)m 22m+1 ] 43 128 ⎛ ⎜ 85 A8 = ⎝ − 128 − 32725 128 85 − 256 . + 1 ,m / 1 3 [ −1 + − 2 . + 1 ,m / 1 3 2 + −2 . + 1 ,m / 1 3 −1 + − 2 171 256 85 − 256 0 0 0 1 m 3 [(−2) + (−1)m 2m+1 ] ⎞ ⎟ ⎠ ⎞ ⎟ 0 ⎠. 256 11. The characteristic equation is λ2 − 8λ + 16 = 0, with eigenvalues 4 and 4. Substituting the eigenvalues into λm = c0 + c1 λ generates 4m = c0 + 4c1 4m−1 m = c1 . Solving the system gives c0 = −4m (m − 1), c1 = 4m−1 m. Thus A m = c0 I + c1 A = ) 4m−1 (3m + 4) 3 · 4m−1 m −3 · 4m−1 m 4m−1 (−3m + 4) A6 = ! 22528 −18432 and 406 18432 −14336 " . * 8.9 Powers of Matrices 12. The characteristic equation is −λ3 − λ2 + 21λ + 45 = 0, with eigenvalues −3, −3, and 5. Substituting the eigenvalues into λm = c0 + c1 λ + c2 λ2 generates (−3)m = c0 − 3c1 + 9c2 (−3)m−1 m = c1 − 6c2 5m = c0 + 5c1 + 25c2 . Solving the system gives 1 [73(−3)m − 2(−1)m 3m+2 + 9 · 5m − 40(−3)m m], 64 1 [−(−1)m 3m+2 + 9 · 5m − 8(−3)m m], c1 = 96 1 [−(−3)m + 5m − 8(−3)m−1 m]. c2 = 64 c0 = Thus Am = c0 I + c1 A + c2 A2 ⎛ 1 m m [31(−3) − (−1) 3m+1 + 4 · 5m ] 1 [−(−3)m 16 − (−1)m 3m+1 + 4 · 5m ] 1 [(−3)m 32 + (−1)m 3m+1 − 4 · 5m ] 1 [−(−3)m 16 − (−1)m 3m+1 + 4 · 5m ] 1 [7(−3)m 8 − (−1)m 3m+1 + 4 · 5m ] 1 [(−3)m 16 3 [(−3)m 32 + (−1)m 3m+1 − 4 · 5m ] 3 [(−3)m 16 + (−1)m 3m+1 − 4 · 5m ] + (−1)m 3m+1 − 4 · 5m ] ⎟ 32 ⎜ =⎜ ⎝ 1 [29(−3)m 32 and ⎛ ⎜ A5 = ⎝ 178 842 842 −1263 1441 −2526 −421 − (−1)m 3m+2 + 12 · 5m ] ⎞ ⎟ ⎠ ⎞ ⎟ −842 ⎠ . 1020 13. (a) The characteristic equation is λ2 − 4λ = λ(λ − 4) = 0, so 0 is an eigenvalue. Since the matrix satisfies the characteristic equation, A2 = 4A, A3 = 4A2 = 42 A, A4 = 42 A2 = 43 A, and, in general, ! " 4m 4m m m A =4 A= . 3(4)m 3(4)m (b) The characteristic equation is λ2 = 0, so 0 is an eigenvalue. Since the matrix satisfies the characteristic equation, A2 = 0, A3 = AA2 = 0, and, in general, Am = 0. (c) The characteristic equation is −λ3 + 5λ2 − 6λ = 0, with eigenvalues 0, 2, and 3. Substituting λ = 0 into λm = c0 + c1 λ + c2 λ2 we find that c0 = 0. Using the nonzero eigenvalues, we find 2m = 2c1 + 4c2 3m = 3c1 + 9c2 . Solving the system gives c1 = 1 [9(2)m − 4(3)m ], 6 c2 = 1 [−3(2)m + 2(3)m ]. 6 3m−1 3m−1 Thus Am = c1 A + c2 A2 and ⎛ ⎜ Am = ⎝ 2(3)m−1 1 m 6 [9(2) − 4(3)m ] 1 m 6 [−9(2) + 8(3)m ] 1 m 6 [3(2) − 2(3)m ] 1 m 6 [−3(2) 407 + 4(3)m ] 1 m 6 [−3(2) 1 m 6 [3(2) ⎞ − 2(3)m ] ⎟ ⎠. + 4(3)m ] 8.9 Powers of Matrices 14. (a) Let Xn−1 = ! xn−1 yn−1 Then Xn = AXn−1 = ! " 1 1 1 0 and A = "! xn−1 " yn−1 = ! ! 1 1 0 1 " . xn−1 + yn−1 xn−1 " . √ √ (b) The characteristic equation of A is λ2 − λ − 1 = 0, with eigenvalues λ1 = 12 (1 − 5 ) and λ2 = 12 (1 + 5 ). m From λm = c0 + c1 λ we get λm 1 = c0 + c1 λ1 and λ2 = c0 + c1 λ2 . Solving this system gives m c0 = (λ2 λm 1 − λ1 λ2 )/(λ2 − λ1 ) m and c1 = (λm 2 − λ1 )/(λ2 − λ1 ). Thus Am = c0 I + c1 A √ √ ! (1 + 5 )m+1 − (1 − 5 )m+1 1 √ √ √ = 2(1 + 5 )m − 2(1 − 5 )m 2m+1 5 √ " 5 )m − 2(1 − 5 )m √ √ m √ √ m . (1 + 5 )(1 − 5 ) − (1 − 5 )(1 + 5 ) 2(1 + √ (c) From part (a), X2 = AX1 , X3 = AX2 = A2 X1 , X4 = AX3 = A3 X1 , and, in general, Xn = An−1 X1 . With ! " "! " ! ! " 1 1 144 89 233 X1 = we have X12 = A11 X1 = = , 89 55 1 1 144 so the number of adult pairs is 233. With ! " ! 1 144 X1 = we have A11 X1 = 89 0 so the number of baby pairs is 144. With ! " ! 2 144 X1 = we have A11 X1 = 89 1 89 55 "! " ! " 1 144 , = 89 0 89 55 "! " ! " 2 377 = , 1 233 so the total number of pairs is 377. 1 A2 + 12 A. Multiplying 15. The characteristic equation of A is λ2 − 5λ + 10 = 0, so A2 − 5A + 10I = 0 and I = − 10 by A−1 we find " ! ! " ) 3 2* 1 1 2 −4 1 1 0 1 10 5 −1 + = A =− A+ I=− . 1 1 10 2 10 1 2 0 1 3 − 10 5 16. The characteristic equation of A is −λ3 +2λ2 +λ−2 = 0, so −A3 +2A2 +A−2I = 0 and I = − 12 A3 +A2 + 12 A. Multiplying by A−1 we find ⎛3 1 2 1 2 1 2 2 1 1 ⎜ A−1 = − A2 + A + I = ⎝ 21 2 2 1 2 − 52 ⎞ ⎟ − 12 ⎠ . − 32 17. (a) Since 2 A = ! 1 −1 0 0 " we see that for all integers m ≥ 2. Thus A is not nilpotent. (b) Since A2 = 0, the matrix is nilpotent with index 2. (c) Since A3 = 0, the matrix is nilpotent with index 3. 408 A m = ! 1 −1 0 0 " 8.10 Orthogonal Matrices (d) Since A2 = 0, the matrix is nilpotent with index 2. (e) Since A4 = 0, the matrix is nilpotent with index 4. (f ) Since A4 = 0, the matrix is nilpotent with index 4. 18. (a) If Am = 0 for some m, then (det A)m = det Am = det 0 = 0, and A is a singular matrix. (b) By (1) of Section 8.8 we have AK = λK, A2 K = λAK = λ2 K, A3 K = λ2 AK = λ3 K, and, in general, Am K = λm K. If A is nilpotent with index m, then Am = 0 and λm = 0. EXERCISES 8.10 Orthogonal Matrices ⎛ ⎞ ⎞ ⎞⎛ ⎞ ⎛ 0 0 0 0 −4 0 ⎜ ⎟ ⎟ ⎟⎜ ⎟ ⎜ ⎜ 0 ⎠ ⎝ 1 ⎠ = ⎝ −4 ⎠ = −4 ⎝ 1 ⎠; λ1 = −4 1. (a)–(b) ⎝ 0 −4 0 0 0 −4 0 15 ⎞⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ 4 0 0 −4 −4 4 ⎟⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ 0 ⎠ ⎝ 0 ⎠ = ⎝ 0 ⎠ = (−1) ⎝ 0 ⎠; λ2 = −1 ⎝ 0 −4 −4 0 15 1 1 1 ⎞ ⎛ ⎞ ⎞ ⎛ ⎞⎛ ⎛ 1 16 1 0 0 −4 ⎟ ⎜ ⎟ ⎟ ⎜ ⎟⎜ ⎜ 0 ⎠ ⎝ 0 ⎠ = ⎝ 0 ⎠ = 16 ⎝ 0 ⎠; λ3 = 16 ⎝ 0 −4 −4 −64 −4 −4 0 15 ⎛ ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ 4 1 1 ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ (c) KT1 K2 = ( 0 1 0 ) ⎝ 0 ⎠ = 0; KT1 K3 = ( 0 1 0 ) ⎝ 0 ⎠ = 0; KT2 K3 = ( 4 0 1 ) ⎝ 0 ⎠ = 0 1 −4 −4 ⎛ 1 −1 ⎜ 2. (a)–(b) ⎝ −1 −1 ⎛ 1 ⎜ ⎝ −1 −1 ⎛ 1 ⎜ −1 ⎝ −1 −1 ⎞⎛ −2 ⎛ 0 ⎞ ⎛ −4 ⎞ ⎛ −2 ⎞ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 1 −1 ⎠ ⎝ 1 ⎠ = ⎝ 2 ⎠ = 2 ⎝ 1 ⎠; λ1 = 2 −1 1 1 2 1 ⎞⎛ ⎞ ⎛ ⎛ ⎞ ⎞ −1 −1 0 0 0 ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 1 −1 ⎠ ⎝ 1 ⎠ = ⎝ 2 ⎠ = 2 ⎝ 1 ⎠; λ2 = 2 −1 1 −1 −2 −1 ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −1 −1 1 −1 1 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 −1 ⎠ ⎝ 1 ⎠ = ⎝ −1 ⎠ = (−1) ⎝ 1 ⎠; λ3 = −1 −1 1 1 −1 1 (c) KT1 K2 = ( −2 1 ⎞ ⎟ ⎜ 1 ) ⎝ 1 ⎠ = 1 − 1 = 0; KT1 K3 = ( −2 −1 409 1 ⎛ ⎞ 1 ⎜ ⎟ 1 ) ⎝ 1 ⎠ = −2 + 1 + 1 = 0 1 8.10 Orthogonal Matrices ⎛ ⎞ 1 ⎜ ⎟ KT2 K3 = ( 0 1 −1 ) ⎝ 1 ⎠ = 1 − 1 = 0 1 ⎛ √2 ⎞ ⎞ ⎛ √2 ⎞ ⎛ √ ⎞ 0 9 2 2 2 ⎜√ ⎟ ⎟⎜ √ ⎟ ⎜ √ ⎟ 0 ⎠ ⎝ 22 ⎠ = ⎝ 9 2 ⎠ = 18 ⎝ 22 ⎠; λ1 = 18 0 0 −8 0 0 ⎛ 5 13 ⎜ 5 3. (a)–(b) ⎝ 13 0 (c) ⎛ 5 13 ⎜ ⎝ 13 0 5 0 ⎛ 5 13 ⎜ ⎝ 13 0 5 0 KT1 K2 =( KT1 K3 = ( KT2 K3 ⎛ =( √ √ √ √ 2 2 2 2 √ 3 2 ⎜ 4. (a)–(b) ⎝ 2 2 ⎛ 3 ⎜ ⎝2 2 ⎛ 3 ⎜ ⎝2 2 2 0 2 2 0 2 2 0 3 3 √ 3 3 √ ⎟⎜ 0 ⎠ ⎝ − 33 √ 3 −8 3 ⎞ ⎛ √6 0 6 ⎟ ⎜ √6 0⎠⎝− 6 √ −8 − 36 0 ⎞⎛ 2 2 2 2 − 3 3 ⎞⎛ √ 8 3 3 √ 8 3 − 3 ⎛ 8√ 6 − 6 ⎜ 8√6 ⎝ 6 √ 8 6 3 ⎞ ⎟ ⎠= √ ⎞ √ ⎞ √ ⎟ ⎠= ⎟ ⎠= √ 6 6 √ 3 ⎜ 6 3 )⎝− 6 √ − 36 √ √ √ −832 ⎟ ⎜ ⎠=⎝ 3 3 √ ⎜ 0 ) ⎝ − 33 √ 3 3 √ ⎛ 6 6 √ ⎜ 0 ) ⎝ − 66 √ − 36 ⎛ ⎛ ⎞ ⎛ ⎞ ⎟ ⎠= ⎞ ⎟ ⎠= √ 3 3 √ ⎜ (−8) ⎝ − 33 √ 3 3 ⎛ √6 6 ⎜ √6 (−8) ⎝ − 6 √ − 36 ⎛ ⎞ ⎟ ⎠; λ2 = −8 ⎞ ⎟ ⎠; λ3 = −8 √ 6 6 − = 0; 6 6 ⎞ √ 12 12 − =0 12 12 ⎟ ⎠= √ √ 18 18 18 + − =0 18 18 9 √ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −2 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎠ ⎝ 2 ⎠ = ⎝ 0 ⎠ = 0 ⎝ 2 ⎠; λ1 = 0 4 1 0 1 ⎞⎛ ⎞ ⎛ ⎛ ⎞ ⎞ 2 1 3 1 ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 0 ⎠ ⎝ 2 ⎠ = ⎝ 6 ⎠ = 3 ⎝ 2 ⎠; λ2 = 3 4 −2 −6 −2 ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎞ 2 12 2 2 ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠ ⎝ 1 ⎠ = ⎝ 6 ⎠ = 6 ⎝ 1 ⎠; λ3 = 6 2 12 2 4 2 −2 ⎛ ⎞ 1 ⎜ ⎟ (c) KT1 K2 = ( −2 2 1 ) ⎝ 2 ⎠ = −2 + 4 − 2 = 0; KT1 K3 = ( −2 −2 ⎛ ⎞ 2 ⎜ ⎟ KT2 K3 = ( 1 2 −2 ) ⎝ 1 ⎠ = 2 + 2 − 4 = 0 2 5. Orthogonal. Columns form an orthonormal set. 410 2 ⎛ ⎞ 2 ⎜ ⎟ 1 ) ⎝ 1 ⎠ = −4 + 2 + 2 = 0 2 8.10 Orthogonal Matrices 6. Not orthogonal. Columns one and three are not unit vectors. 7. Orthogonal. Columns form an orthonormal set. 8. Not orthogonal. The matrix is singular. 9. Not orthogonal. Columns are not unit vectors. 10. Orthogonal. Columns form an orthogonal set. ) 1 " ! " ! √ 1 1 2 11. λ1 = −8, λ2 = 10, K1 = , K2 = , P= −1 1 − √12 * ) ! " ! " 1 0 1 0 12. λ1 = 7, λ2 = 4, K1 = , K2 = , P= 0 1 0 1 ) " ! " ! √3 1 3 10 13. λ1 = 0, λ2 = 10, K1 = , K2 = , P= 1 −1 3 √ − 10 √ √ 1 1 5 5 14. λ1 = + , λ2 = − , K1 = 2 2 2 2 ! √1 2 √1 2 * √1 10 √3 10 * √ ⎛ 1+ 5 √ " √ " ! √ √ 1+ 5 1− 5 10+2 5 ⎝ , K2 = , P= 2 2 √ 2 √ 10+2 5 √ 5 √ 10−2 5 √ 2 √ 10−2 5 √1− ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 − √2 √12 0 −1 1 0 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ 15. λ1 = 0, λ2 = 2, λ3 = 1, K1 = ⎝ 0 ⎠, K2 = ⎝ 0 ⎠, K3 = ⎝ 1 ⎠, P = ⎝ 0 0 1⎠ √1 √1 1 1 0 0 2 2 ⎛ ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ − √12 1 1 −1 ⎜ √ √ √ √ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 16. λ1 = −1, λ2 = 1− 2 , λ3 = 1+ 2 , K1 = ⎝ 0 ⎠, K2 = ⎝ − 2 ⎠, K3 = ⎝ 2 ⎠, P = ⎜ ⎝ 0 1 1 1 √1 ⎛ ⎞ ⎛ ⎞ ⎛ √3 − 11 −3 1 1 ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ √1 17. λ1 = −11, λ2 = 0, λ3 = 6, K1 = ⎝ 1 ⎠, K2 = ⎝ −4 ⎠, K3 = ⎝ 2 ⎠, P = ⎝ 11 1 7 1 √1 ⎛ ⎛ ⎞ ⎞ 11 2 √1 66 − √466 √7 66 ⎞ ⎠ − 1 2 √ 2 2 1 2 √1 6 √2 6 √1 6 ⎞ ⎟ ⎠ 1 2 √ 2 2 1 2 ⎞ ⎟ ⎟ ⎠ ⎞ ⎛ ⎞ ⎛ 1 − 23 32 1 −2 2 3 ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ 1 2 ⎟ 18. λ1 = −18, λ2 = 0, λ3 = 9, K1 = ⎝ −2 ⎠, K2 = ⎝ 1 ⎠, K3 = ⎝ 2 ⎠, P = ⎝ − 23 3 3 ⎠ 2 2 1 2 2 1 3 3 3 *! )3 ! " 3 4 " a 1 0 5 5 5 9 4 3 2 = implies 25 + a2 = 1 and 16 19. 25 + b = 1. These equations give a = ± 5 , b = ± 5 . 4 a b 0 1 b 5 ⎛ ⎛ ⎞ ⎞ + ab = 0 indicates a and b must have opposite signs. Therefore choose a = − 54 , b = 35 . )3 * − 45 5 The matrix 4 is orthogonal. 3 But 12 25 5 20. ) √1 5 b a √1 5 *! 5 √1 5 a b √1 5 " = ! 1 0 0 1 " implies 1 5 + b2 = 1 and a2 + 1 5 = 1. These give a = ± √25 , b = ± √25 . a b But √ + √ = 0 indicates a and b must have opposite signs. Therefore choose a = − √25 , b = 5 5 * ) 1 2 The matrix √ 5 2 √ − 5 √ 5 √1 5 is orthogonal. 411 √2 2 . 8.10 Orthogonal Matrices 21. (a)–(b) We compute ⎛ 0 2 ⎜ AK1 = ⎝ 2 0 2 ⎛ 0 ⎜ AK2 = ⎝ 2 2 ⎛ 0 ⎜ AK3 = ⎝ 2 2 2 ⎞⎛ 1 ⎛ ⎞ −2 ⎛ ⎞ 1 ⎞ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 2 ⎠ ⎝ −1 ⎠ = ⎝ 2 ⎠ = −2 ⎝ −1 ⎠ = −2K1 2 0 0 0 0 ⎛ ⎞ ⎞ ⎞⎛ ⎞ ⎛ 1 1 −2 2 2 ⎜ ⎟ ⎟ ⎟⎜ ⎟ ⎜ 0 2 ⎠ ⎝ 0 ⎠ = ⎝ 0 ⎠ = −2 ⎝ 0 ⎠ = −2K2 2 −1 2 0 −1 ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 2 1 4 1 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 2 ⎠ ⎝ 1 ⎠ = ⎝ 4 ⎠ = 4 ⎝ 1 ⎠ = 4K3 , 2 0 1 4 1 and observe that K1 is an eigenvector with corresponding eigenvalue −2, K2 is an eigenvector with corresponding eigenvalue −2, and K3 is an eigenvector with corresponding eigenvalue 4. (c) Since K1 · K2 = 1 ̸= 0, K1 and K2 are not orthogonal, while K1 · K3 = 0 and K2 · K3 = 0 so K3 is orthogonal to both K1 and K2 , To transform {K1 , K2 } into an orthogonal set we let V1 = K1 and compute K2 · V1 = 1 and V1 · V1 = 2. Then ⎞ ⎛ 1⎞ 1 2 K2 · V 1 ⎟ 1⎜ ⎜ ⎟ ⎜ ⎟ V1 = ⎝ 0 ⎠ − ⎝ −1 ⎠ = ⎝ 12 ⎠ . V 2 = K2 − V1 · V 1 2 −1 0 −1 ⎛ 1 ⎞ ⎛ Now, {V1 , V2 , K3 } is an orthogonal set of eigenvectors with ||V1 || = √ 2, 3 ||V2 || = √ , 6 and ||K3 || = An orthonormal set of vectors is √1 2 ⎜ √1 ⎝− 2 ⎛ 0 ⎞ ⎟ ⎠, √1 6 ⎜ √1 ⎝ 6 − √26 ⎛ ⎞ ⎟ ⎠, and ⎛ ⎜ ⎝ and so the matrix P= √1 2 ⎜ √1 ⎝− 2 ⎛ 0 √1 6 √1 6 − √26 is orthogonal. 412 √1 3 √1 3 √1 3 ⎞ ⎟ ⎠ √1 3 √1 3 √1 3 ⎞ ⎟ ⎠, √ 3. 8.10 Orthogonal Matrices 22. (a)–(b) We compute 1 1 1 1 ⎜1 ⎜ AK1 = ⎜ ⎝1 1 1 1 1 1 1 1 1 ⎜1 ⎜ AK2 = ⎜ ⎝1 1 ⎛ 1 ⎜1 ⎜ AK3 = ⎜ ⎝1 1 ⎛ 1 ⎜1 ⎜ AK4 = ⎜ ⎝1 1 1 1 1 1 1 1 1 1 ⎜ 1⎟ ⎟⎜ ⎟⎜ 1⎠⎝ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎛ ⎛ ⎞⎛ −1 ⎛ ⎞ ⎛ ⎞ 0 −1 ⎜ ⎟ ⎜ 0⎟ 0⎟ ⎟ ⎜0⎟ ⎜ ⎟ ⎟ = ⎜ ⎟ = 0⎜ ⎟ = 0K1 ⎝ 0⎠ 0⎠ ⎝0⎠ 1 ⎞ 1 0 1 ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −1 0 −1 ⎜ ⎟ ⎜ ⎟ ⎜ 0⎟ 1⎟ ⎟⎜ 0⎟ ⎜0⎟ ⎜ ⎟ ⎟⎜ ⎟ = ⎜ ⎟ = 0⎜ ⎟ = 0K2 ⎝ 1⎠ 1⎠⎝ 1⎠ ⎝0⎠ 1 0 0 0 ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −1 0 −1 ⎜ ⎟ ⎜ ⎟ ⎜ 1⎟ 1⎟ ⎟⎜ 1⎟ ⎜0⎟ ⎜ ⎟ ⎟⎜ ⎟ = ⎜ ⎟ = 0⎜ ⎟ = 0K3 ⎝ 0⎠ 1⎠⎝ 0⎠ ⎝0⎠ 1 0 0 0 ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎞ 1 4 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1⎟⎜1⎟ ⎜4⎟ ⎜1⎟ ⎟ ⎜ ⎟ = ⎜ ⎟ = 4 ⎜ ⎟ = 4K4 , ⎝1⎠ 1⎠⎝1⎠ ⎝4⎠ 1 1 4 1 and observe that K1 is an eigenvector with corresponding eigenvalue 0, K2 is an eigenvector with corresponding eigenvalue 0, K3 is an eigenvector with corresponding eigenvalue 0, and K4 is an eigenvector with corresponding eigenvalue 4. (c) Since K1 · K2 = 1 ̸= 0, K1 and K2 are not orthogonal. Similarly , K1 · K3 = 1 ̸= 0 and K2 · K3 = 1 ̸= 0 so K1 and K3 and K2 and K3 are not orthogonal. However, K1 · K4 = 0, K2 · K4 = 0, and K3 · K4 = 0, so each of K1 , K2 , and K3 is orthogonal to K4 . To transform {K1 , K2 , K3 } into an orthogonal set we let V1 = K1 and compute K2 · V1 = 1 and V1 · V1 = 2. Then ⎞ ⎛ ⎛ ⎞ ⎛ 1⎞ −2 −1 −1 ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ K2 · V1 ⎜ 0⎟ 1 ⎜ 0⎟ ⎜ 0⎟ V1 = ⎜ V2 = K2 − ⎟− ⎜ ⎟. ⎟=⎜ V1 · V1 ⎝ 1⎠ 2 ⎝ 0⎠ ⎝ 1⎠ 0 − 12 1 Next, using K3 · V1 = 1, K3 · V2 = 21 , and V2 · V2 = 32 , we obtain ⎛ ⎛ ⎛ 1⎞ ⎛ 1⎞ ⎞ ⎞ −3 −1 −1 −2 ⎜ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ K3 · V 1 K3 · V2 ⎜ 1 ⎟ 1 ⎜ 0 ⎟ 1/2 ⎜ 0 ⎟ ⎜ 1 ⎟ V3 = K 3 = V1 − V2 = ⎜ ⎟ = ⎜ 1 ⎟. ⎜ ⎟− ⎟− ⎜ V1 · V1 V2 · V2 ⎝ 0 ⎠ 2 ⎝ 0 ⎠ 3/2 ⎝ 1 ⎠ ⎝ − 3 ⎠ 0 − 12 1 Now, {V1 , V2 , V3 , K4 } is an orthogonal set of eigenvectors with √ 3 2 ||V1 || = 2, ||V2 || = √ , ||K3 || = √ and ||K4 ∥ = 2. 6 3 An orthonormal set of vectors is ⎛ 1 ⎞ − √2 ⎜ 0 ⎟ ⎟ ⎜ ⎟, ⎜ ⎝ 0 ⎠ √1 2 ⎞ − √16 ⎜ 0 ⎟ ⎜ ⎟ ⎜ 2 ⎟, ⎝ √6 ⎠ ⎛ − √16 ⎛ 1 − 2√ 3 ⎞ 3 ⎟ ⎜ √ ⎜ 2 3⎟ ⎟, ⎜ 1 ⎠ ⎝− √ 2 3 1 − 2√ 3 413 ⎛1⎞ 2 and ⎜1⎟ ⎜2⎟ ⎜ 1 ⎟, ⎝2⎠ 1 2 − 13 8.10 Orthogonal Matrices and so the matrix ⎛ ⎜ ⎜ P=⎜ ⎝ − √16 − √12 0 0 √2 6 − √16 0 √1 2 1 − 2√ 3 3 √ 2 3 1 − 2√ 3 1 − 2√ 3 1 2 1 2 1 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠ is orthogonal. ⎛ ⎞ ⎛ ⎞ a 0 ⎜ ⎟ ⎜ ⎟ 23. If we take K1 = ⎝ 1 ⎠ as in Example 4 in the text then we look for a vector K2 = ⎝ b ⎠ such that 1(a) + c 1 1 1 1 1 1 4 b − 4 c = 0 and K1 · K2 = 0 or b + c = 0. The last equation implies c = −b so a + 4 b − 4 (−b) = a + 2 b = 0. If we let ⎛ b =⎞−2, then a = 1 and c = 2, so a second eigenvector with eigenvalue −9 and orthogonal to K1 is 1 ⎟ ⎜ K2 = ⎝ −2 ⎠. 2 24. The eigenvalues and corresponding eigenvectors of A are ⎞ ⎛ −1 ⎜ 1⎟ ⎟ ⎜ λ1 = λ2 = −1, λ3 = λ4 = 3, and K1 = ⎜ ⎟, ⎝ 0⎠ 0 ⎞ 0 ⎜ 0⎟ ⎟ ⎜ K2 = ⎜ ⎟, ⎝ −1 ⎠ ⎛ 1 ⎛ ⎞ 0 ⎜0⎟ ⎜ ⎟ K3 = ⎜ ⎟ , ⎝1⎠ 1 ⎛ ⎞ 1 ⎜1⎟ ⎜ ⎟ K4 = ⎜ ⎟ . ⎝0⎠ 0 Since K1 · K2 = K1 · K3 = K1 · K4 = K2 · K3 = K2 · K4 = K3 · K4 = 0, the vectors are orthogonal. Using √ 2 , we construct the orthogonal matrix ⎞ ⎛ √1 √1 0 0 − 2 2 ⎟ ⎜ √1 √1 ⎟ 0 0 ⎜ 2 2 ⎟ P=⎜ ⎟. ⎜ 0 ⎠ ⎝ 0 − √12 √12 ∥K1 ∥ = ∥K2 ∥ = ∥K3 ∥ = ∥K4 ∥ = 0 √1 2 √1 2 0 25. Suppose A and B are orthogonal matrices. Then A−1 = AT and B−1 = BT and (AB)−1 = B−1 A−1 = BT AT = (AB)T . Thus AB is an orthogonal matrix. EXERCISES 8.11 Approximation of Eigenvalues ! " 1 and computing Xi = AXi−1 for i = 1, 2, 3, 4 we obtain 1 ! " ! " ! " ! " 2 4 8 16 X1 = , X2 = , X3 = , X4 = . 2 4 8 16 ! " 1 AK · K 4 with corresponding eigenvalue λ = = = 2. We conclude that a dominant eigenvector is K = K·K 2 1 1. Taking X0 = 414 8.11 Approximation of Eigenvalues ! " 1 and computing Xi = AXi−1 for i = 1, 2, 3, 4, 5 we obtain 1 ! " ! " ! " ! " ! " −5 −437 49 −35429 3937 X1 = , X3 = , X2 = , X5 = , X4 = . −47 7 −3935 439 35431 ! " " ! −35429 −0.99994 1 with corresponding ≈ We conclude that a dominant eigenvector is K = 35439 1 35431 AK · K = −8.9998. eigenvalue λ = K·K ! " ! " ! " ! " 6 1 6 0.375 1 3. Taking X0 = , we define X1 = and computing AX0 = = . Continuing in this 16 16 16 1 1 2. Taking X0 = manner we obtain " " ! 0.3333 0.3335 . , X4 = 1 1 " ! 0.3333 with corresponding eigenvalue λ = 14. We conclude that a dominant eigenvector is K = 1 ! " ! " ! " ! " 1 0.2 1 1 1 4. Taking X0 = . Continuing in this manner = and computing AX0 = , we define X1 = 5 5 1 1 5 we obtain " " ! " ! " ! ! 0.2679 0.2680 0.2676 0.2727 . , X5 = , X4 = , X3 = X2 = 1 1 1 1 " ! 0.2679 with corresponding eigenvalue λ = 6.4641. We conclude that a dominant eigenvector is K = 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 11 1 11 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 5. Taking X0 = ⎝ 1 ⎠ and computing AX0 = ⎝ 11 ⎠, we define X1 = ⎝ 11 ⎠ = ⎝ 1 ⎠. Continuing in 11 6 0.5455 6 1 X2 = ! 0.3363 1 " , X3 = ! this manner we obtain ⎛ ⎜ X2 = ⎝ 1 1 0.5045 ⎞ ⎟ ⎠, ⎛ ⎜ X3 = ⎝ ⎛ 1 1 0.5005 ⎞ ⎞ ⎟ ⎠, ⎛ ⎞ 1 ⎜ ⎟ X4 = ⎝ 1 ⎠ . 0.5 1 ⎜ ⎟ We conclude that a dominant eigenvector is K = ⎝ 1 ⎠ with corresponding eigenvalue λ = 10. 0.5 ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ 5 1 1 5 1⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ 6. Taking X0 = ⎝ 1 ⎠ and computing AX0 = ⎝ 2 ⎠, we define X1 = ⎝ 2 ⎠ = ⎝ 0.4 ⎠. Continuing in this 5 0.4 2 1 2 manner we obtain ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X2 = ⎝ 0.2105 ⎠ , X3 = ⎝ 0.1231 ⎠ , X4 = ⎝ 0.0758 ⎠ , X5 = ⎝ 0.0481 ⎠ . 0.0481 0.0758 0.1231 ⎛ ⎞ ⎛ ⎞ 1 1 ⎜ ⎟ ⎜ ⎟ At this point if we restart with X0 = ⎝ 0 ⎠ we see that K = ⎝ 0 ⎠ is a dominant eigenvector with corresponding 0 0 eigenvalue λ = 3. 0.2105 415 8.11 Approximation of Eigenvalues ! " 1 and using scaling we obtain 1 " " ! " ! " ! " ! ! 0.5008 0.5028 0.5098 0.5345 0.625 . , X5 = , X4 = , X3 = , X2 = X1 = 1 1 1 1 1 ! " 0.5 Taking K = as the dominant eigenvector we find λ1 = 7. Now the normalized eigenvector is 1 " ! " " ! ! 1 1.6 −0.8 0.4472 K1 = . Taking X0 = and B = and using scaling again we obtain X1 = −0.8 0.4 0.8944 1 " ! ! " ! " 1 1 1 . Taking K = we find λ2 = 2. The eigenvalues are 7 and 2. , X2 = −0.5 −0.5 −0.5 7. Taking X0 = 8. Taking X0 = ! " " " ! " ! ! 1 1/3 0.3333 0.3333 as the . Taking K = , X2 = and using scaling we obtain X1 = 1 1 1 1 dominant eigenvector we find λ1 = 10. Now the normalized eigenvector is K1 = ! 0.3162 0.9486 An eigenvector for the zero matrix is λ2 = 0. The eigenvalues are 10 and 0. ⎛ ⎞ 1 ⎜ ⎟ 9. Taking X0 = ⎝ 1 ⎠ and using scaling we obtain 1 ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 1 1 1 ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X1 = ⎝ 0 ⎠ , X2 = ⎝ −0.6667 ⎠ , X3 = ⎝ −0.9091 ⎠ , X4 = ⎝ −0.9767 ⎠ , ⎛ 1 1 1 1 ⎞ " and B = ⎛ 1 ! 0 0 " 0 . 0 ⎞ ⎟ ⎜ X5 = ⎝ −0.9942 ⎠ . 1 1 ⎜ ⎟ Taking K = ⎝ −1 ⎠ as the dominant eigenvector we find λ1 = 4. Now the normalized eigenvector is 1 ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ 1 1.6667 0.3333 −1.3333 0.5774 ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ 0.3333 ⎠. If X0 = ⎝ 1 ⎠ is now chosen only one more K1 = ⎝ −0.5774 ⎠and B = ⎝ 0.3333 0.6667 1 −1.3333 0.3333 1.6667 0.5774 ⎛ ⎞ 1 ⎜ ⎟ eigenvalue is found. Thus, try X0 = ⎝ 1 ⎠. Using scaling we obtain 0 ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X1 = ⎝ 0.5 ⎠ , X2 = ⎝ 0.2 ⎠ , X3 = ⎝ 0.0714 ⎠ , X4 = ⎝ 0.0244 ⎠ , X5 = ⎝ 0.0082 ⎠ . −0.9918 −0.9756 −0.9286 −0.8 −0.5 ⎛ ⎞ 1 ⎜ ⎟ Taking K = ⎝ 0 ⎠ as the eigenvector we find λ2 = 3. The normalized eigenvector in this case is −1 ⎞ 0.7071 ⎟ ⎜ K2 = ⎝ 0 ⎠ and −0.7071 ⎛ ⎛ 0.1667 0.3333 ⎜ C = ⎝ 0.3333 0.1667 0.6667 0.3333 ⎛ ⎞ 1 ⎜ ⎟ ⎟ 0.3333 ⎠. If X0 = ⎝ 1 ⎠ is chosen, and scaling is used we 0.1667 1 0.1667 416 ⎞ 8.11 Approximation of Eigenvalues ⎛ 0.5 ⎞ ⎛ 0.5 ⎛ ⎞ 0.5 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ obtain X1 = ⎝ 1 ⎠, X2 = ⎝ 1 ⎠. Taking K = ⎝ 1 ⎠ we find λ3 = 1. The eigenvalues are 4, 3, and 1. 0.5 0.5 0.5 ⎛ ⎞ 1 ⎜ ⎟ The difficulty in choosing X0 = ⎝ 1 ⎠ to find the second eigenvector results from the fact that this vector is a 1 linear combination of the eigenvectors corresponding to the other two eigenvalues, with 0 contribution from the second eigenvector. When this occurs the development of the power method, shown in the text, breaks down. ⎛ ⎞ 1 ⎜ ⎟ 10. Taking X0 = ⎝ 1 ⎠ and using scaling we obtain 1 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ −0.2499 −0.2504 −0.2431 −0.3636 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X1 = ⎝ −0.3636 ⎠ , X2 = ⎝ 0.0884 ⎠ , X3 = ⎝ −0.0221 ⎠ , X4 = ⎝ −0.0055 ⎠ . ⎛ ⎜ Taking K = ⎝ −0.25 0 1 ⎞ ⎟ ⎠ as the dominant eigenvector we find λ1 = 16. The normalized eigenvector is ⎛ −0.9412 −0.2425 ⎜ ⎟ ⎜ 0 0 K1 = ⎝ ⎠ and B = ⎝ ⎛ −0.2353 0.9701 ⎛ ⎜ X1 = ⎝ 1 1 1 1 ⎞ −0.2941 ⎞ ⎟ −1 ⎠ , −0.0735 0 −4 0 ⎛ ⎜ X2 = ⎝ ⎛ ⎞ ⎞ 1 −0.2353 ⎜ ⎟ ⎟ 0 ⎠. Taking X0 = ⎝ 1 ⎠ and using scaling we obtain 1 −0.0588 0.0735 ⎞ ⎟ 1 ⎠, 0.0184 ⎛ ⎜ X3 = ⎝ −0.0184 ⎞ ⎟ −1 ⎠ , −0.0046 ⎛ ⎜ X4 = ⎝ 0.0046 ⎞ ⎟ 1 ⎠. 0.0011 ⎛ ⎞ 0 ⎜ ⎟ Taking K = ⎝ 1 ⎠ as the eigenvector we find λ2 = −4. The normalized eigenvector in this case is K2 = K = 0 ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ ⎞ ⎛ 0 −0.9412 0 −0.2353 1 −1 ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ 0 0 0 ⎝ 1 ⎠ and C = ⎝ ⎠. Taking X0 = ⎝ 1 ⎠ and using scaling we obtain X1 = ⎝ 0 ⎠, −0.2353 0 −0.0588 0 1 −0.25 ⎞ ⎛ ⎞ ⎛ 1 1 ⎟ ⎜ ⎟ ⎜ X2 = ⎝ 0 ⎠. Using K = ⎝ 0 ⎠ we find λ3 = −1. The eigenvalues are 16, −4, and −1. 0.25 " ! " 1 4 −1 and using scaling we obtain . Taking X0 = 11. The inverse matrix is −3 1 1 " " ! " ! " ! ! 1 1 1 1 . , X4 = , X3 = , X2 = X1 = −0.7913 −0.7910 −0.7857 −0.6667 " ! ! " 1 1 1 we find λ = 4.7913. The minimum eigenvalue of Using K = is 1/4.7913 ≈ 0.2087. −0.7913 3 4 0.25 ! 417 8.11 Approximation of Eigenvalues ! 12. The inverse matrix is 1 3 4 2 " . Taking X0 = X1 = Using K = ! 0.75 1 " ! 0.6667 1 " , ! " 1 1 X2 = and using scaling we obtain ! 0.7857 1 " 0.75 , . . . , X10 = ! −0.2 0.3 we find λ = 5. The minimum eigenvalue of ! 0.4 −0.1 1 " " . is 1/5 = 0.2 13. (a) Replacing the second derivative with the difference expression we obtain EI yi+1 − 2yi + yi−1 + P yi = 0 or EI(yi+1 − 2yi + yi−1 ) + P h2 yi = 0. h2 (b) Expanding the difference equation for i = 1, 2, 3 and using h = L/4, y0 = 0, and y4 = 0 we obtain EI(y2 − 2y1 + y0 ) + P L2 y1 = 0 16 EI(y3 − 2y2 + y1 ) + P L2 y2 = 0 16 EI(y4 − 2y3 + y2 ) + P L2 y3 = 0 16 or 2y1 − y2 = P L2 y1 16EI −y1 + 2y2 − y3 = P L2 y2 16EI −y2 + 2y3 = P L2 y3 . 16EI In matrix form this becomes ⎛ 2 ⎜ ⎝ −1 0 ⎛ 0.75 0.5 0.25 ⎜ (c) A−1 = ⎝ 0.5 0.25 ⎞⎛ ⎞ ⎛ ⎞ y1 −1 0 y1 P L2 ⎜ ⎟ ⎟⎜ ⎟ 2 −1 ⎠ ⎝ y2 ⎠ = ⎝ y2 ⎠ . 16EI −1 2 y3 y3 ⎞ ⎟ 1 0.5 ⎠ 0.5 0.75 ⎛ ⎞ 1 ⎜ ⎟ (d) Taking X0 = ⎝ 1 ⎠ and using scaling we obtain 1 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 0.75 0.7143 0.7083 0.7073 0.7071 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X1 = ⎝ 1 ⎠ , X2 = ⎝ 1 ⎠ , X3 = ⎝ 1 ⎠ , X4 = ⎝ 1 ⎠ , X5 = ⎝ 1 ⎠ . 0.75 0.7143 0.7083 0.7073 0.7071 ⎞ ⎛ 0.7071 ⎟ ⎜ Using K = ⎝ 1 ⎠ we find λ = 1.7071. Then 1/λ = 0.5859 is the minimum eigenvalue of A. 0.7071 (e) Solving EI P L2 = 0.5859 for P we obtain P = 9.3726 2 . In Example 3 of Section 3.9 we saw 16EI L P = π2 EI EI ≈ 9.8696 2 . 2 L L 14. (a) The difference equation is EIi (yi+1 − 2yi + yi−1 ) + P h2 yi = 0, i = 1, 2, 3, 418 8.11 Approximation of Eigenvalues where I0 = 0.00200, I1 = 0.00175, I2 = 0.00150, I3 = 0.00125, and I4 = 0.00100. The system of equations is P L2 P L2 y1 = 0 y1 0.0035y1 − 0.00175y2 = 16 16E P L2 P L2 or y2 = 0 y2 −0.0015y1 + 0.003y2 − 0.0015y3 = 0.00150E(y3 − 2y2 + y1 ) + 16 16E P L2 P L2 0.00125E(y4 − 2y3 + y2 ) + −0.00125y2 + 0.0025y3 = y2 = 0 y3 . 16 16E In matrix form this becomes ⎞⎛ ⎞ ⎛ ⎛ ⎞ 0.0035 −0.00175 0 y1 y1 ⎟ ⎜ ⎟ P L2 ⎜ ⎟ ⎜ 0.003 −0.0015 ⎠ ⎝ y2 ⎠ = ⎝ y2 ⎠ . ⎝ −0.0015 16E y3 0 −0.00125 0.0025 y3 0.00175E(y2 − 2y1 + y0 ) + (b) The inverse of A is A−1 ⎛ 428.571 ⎜ = ⎝ 285.714 142.857 333.333 666.667 333.333 ⎛ ⎞ 1 ⎜ ⎟ Taking X0 = ⎝ 1 ⎠ and using scaling we obtain 1 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 0.6645 0.6710 0.7113 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ X1 = ⎝ 1 ⎠ , X2 = ⎝ 1 ⎠ , X3 = ⎝ 1 ⎠ , 0.7635 0.7679 0.7958 ⎞ 200 ⎟ 400 ⎠ . 600 ⎞ 0.6634 ⎟ ⎜ X4 = ⎝ 1 ⎠ , 0.7628 ⎛ ⎞ 0.6632 ⎟ ⎜ X5 = ⎝ 1 ⎠ . 0.7627 ⎛ This yields the eigenvalue λ = 1161.23. The smallest eigenvalue of A is then 1/λ = 0.0008612. The lowest critical load is 16E E P = (0.0008612) − 0.01378 2 . L2 L 15. (a) A10 (b) X10 X12 ⎞ 67,745,349 −43,691,832 8,258,598 ⎟ ⎜ 28,182,816 −5,328,720 ⎠ = ⎝ −43,691,832 ⎛ 8,258,598 −5,328,720 1,008,180 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 67,745,349 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = A10 ⎝ 0 ⎠ = ⎝ −43,691,832 ⎠ ≈ 67,745,349 ⎝ −0.644942 ⎠ 0.121906 8,258,598 0 ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ 1 2,680,201,629 1 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ = A12 ⎝ 0 ⎠ = ⎝ −1,728,645,624 ⎠ ≈ 2,680,201,629 ⎝ −0.644968 ⎠ . 0 326,775,222 0.121922 The vectors appear to be approaching scalar multiples of K = (1, −0.644968, 0.121922), which approximates the dominant eigenvector. (c) The dominant eigenvalue is λ1 = (AK · K)/(K · K) = 6.28995. 419 8.12 8.11 Diagonalization Approximation of Eigenvalues EXERCISES 8.12 Diagonalization 1. Distinct eigenvalues λ1 = 1, λ2 = 5 imply A is diagonalizable. ! " ! −3 1 1 P= , D= 1 1 0 0 5 2. Distinct eigenvalues λ1 = 0, λ2 = 6 imply A is diagonalizable. " ! ! 0 −5 −1 , D= P= 4 2 0 ! " 1 3. For λ1 = λ2 = 1 we obtain the single eigenvector K1 = . Hence 1 √ √ 4. Distinct eigenvalues λ1 = 5 , λ2 = − 5 imply A is diagonalizable. √ " !√ !√ 5 − 5 5 P= , D= 1 1 0 5. Distinct eigenvalues λ1 = −7, λ2 = 4 imply A is diagonalizable. " ! ! −7 13 1 , D= P= 2 1 0 " 0 6 " A is not diagonalizable. 0 √ − 5 0 4 " " 6. Distinct eigenvalues λ1 = −4, λ2 = 10 imply A is diagonalizable. " " ! ! −4 0 −3 1 , D= P= 0 10 1 −5 1 2 , λ2 = imply A is diagonalizable. 3 3 " !1 ! " 1 1 0 3 , D= P= −1 1 0 23 ! " 1 8. For λ1 = λ2 = −3 we obtain the single eigenvector K1 = . Hence A is not diagonalizable. 1 9. Distinct eigenvalues λ1 = −i, λ2 = i imply A is diagonalizable. " " ! ! −i 0 1 1 , D= P= 0 i −i i 7. Distinct eigenvalues λ1 = 10. Distinct eigenvalues λ1 = 1 + i, λ2 = 1 − i imply A is diagonalizable. ! " ! " 2 2 1+i 0 P= , D= i −i 0 1−i 11. Distinct eigenvalues λ1 = 1, λ2 = −1, λ3 = 2 ⎛ 1 ⎜ P = ⎝0 0 imply A is diagonalizable. ⎛ ⎞ ⎞ 1 0 0 0 1 ⎜ ⎟ ⎟ 1 1 ⎠ , D = ⎝ 0 −1 0 ⎠ 0 1 0 420 0 2 8.12 12. Distinct eigenvalues λ1 = 3, λ2 = 4, λ3 = 5 imply A is diagonalizable. ⎛ ⎛ ⎞ 1 2 0 3 0 ⎜ ⎜ ⎟ 1⎠, D = ⎝0 4 P = ⎝0 2 1 1 −1 0 0 13. Distinct eigenvalues λ1 = 0, λ2 = 1, λ3 = 2 imply ⎛ 1 1 ⎜ P=⎝ 0 1 −1 1 A is diagonalizable. ⎛ ⎞ 1 0 0 ⎜ ⎟ 0⎠, D = ⎝0 1 1 0 0 0 Diagonalization ⎞ ⎟ 0⎠ 5 ⎞ 0 ⎟ 0⎠ 2 14. Distinct eigenvalues λ1 = 1, λ2 = −3i, λ3 = 3i imply A is diagonalizable. ⎛ ⎞ ⎛ ⎞ 1 0 0 0 −3i 3i ⎜ ⎟ ⎜ ⎟ 1 1 ⎠ , D = ⎝ 0 −3i 0 ⎠ P = ⎝0 1 0 0 0 0 3i ⎛ ⎞ 1 ⎜ ⎟ 15. The eigenvalues are λ1 = λ2 = 1, λ3 = 2. For λ1 = λ2 = 1 we obtain the single eigenvector K1 = ⎝ 0 ⎠. 0 Hence A is not diagonalizable. 16. Distinct eigenvalues λ1 = 1, λ2 = 2, λ3 = 3 imply A is diagonalizable. ⎛ ⎛ ⎞ ⎞ 1 1 0 1 0 0 ⎜ ⎜ ⎟ ⎟ P = ⎝0 1 0⎠, D = ⎝0 2 0⎠ 0 0 1 0 0 3 √ √ 17. Distinct eigenvalues λ1 = 1, λ2 = 5 , λ3 = − 5 imply A is diagonalizable. √ √ ⎞ ⎛ ⎛ ⎞ 0 1+ 5 1− 5 1 0 0 √ ⎜ ⎜ ⎟ ⎟ P = ⎝0 5 0⎠ 2 2 ⎠, D = ⎝0 √ 0 0 − 5 1 0 0 ⎛ ⎞ 1 ⎜ ⎟ 18. For λ1 = λ2 = λ3 = 1 we obtain the single eigenvector K1 = ⎝ −2 ⎠. Hence A is not diagonalizable. 1 19. For the eigenvalues λ1 = λ2 = 2, λ3 = 1, λ4 = −1 we obtain four linearly independent eigenvectors. Hence A is diagonalizable and ⎞ ⎞ ⎛ ⎛ 2 0 0 0 −3 −1 −1 1 ⎜0 2 0 ⎜ 0 1 0 0⎟ 0⎟ ⎟ ⎟ ⎜ ⎜ P=⎜ ⎟. ⎟, D = ⎜ ⎝0 0 1 ⎝ −3 0 0 1⎠ 0⎠ 0 0 0 −1 1 0 1 0 ⎛ ⎞ 1 ⎜0⎟ ⎜ ⎟ 20. The eigenvalues are λ1 = λ2 = 2, λ3 = λ4 = 3. For λ3 = λ4 = 3 we obtain the single eigenvector K1 = ⎜ ⎟. ⎝1⎠ 0 Hence A is not diagonalizable. ) 1 " ! " ! √ 1 1 2 21. λ1 = 0, λ2 = 2, K1 = , K2 = , P= 1 −1 1 √ − 2 421 √1 2 √1 2 * , D= ! 0 0 0 2 " 8.12 Diagonalization 22. λ1 = −1, λ2 = 4, K1 = ! ) 1 " ! " √ 2 1 5 , K2 = , P= −2 1 − √2 √2 5 √1 5 5 * , D= ) √10 !√ " ! √ " − √14 10 − 10 , K2 = , P= 23. λ1 = 3, λ2 = 10, K1 = 2 5 √2 √ √10 35 √15 35 14 24. λ1 = −1, λ2 = 3, K1 = ! " 1 1 , K2 = ! ) 1 " √ 1 2 , P= −1 √1 2 √1 2 − √12 −1 0 0 4 , D= ! −1 0 0 3 ! * * , D= ! √1 2 √1 2 ⎛ ⎞ ⎛ ⎞ ⎛ 1 − √2 1 0 ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ 1 25. λ1 = −1, λ2 = λ3 = 1, K1 = ⎝ 1 ⎠, K2 = ⎝ 1 ⎠, K3 = ⎝ 0 ⎠, P = ⎝ √ 2 0 0 1 0 ⎛ −1 ⎞ 0 ⎛ ⎞ ⎛ ⎛ √1 ⎞ − 2 1 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ 0 26. λ1 = λ2 = −1, λ3 = 5, K1 = ⎝ 0 ⎠, K2 = ⎝ 1 ⎠, K3 = ⎝ −1 ⎠, P = ⎝ 1 0 1 √1 2 ⎛ ⎞ −1 0 0 ⎜ ⎟ D = ⎝ 0 −1 0 ⎠ ⎛ 0 −1 ⎞ " 3 0 0 10 " " 0 ⎛ ⎞ 2 3 − 31 − 32 3 0 1 0 ⎟ 0⎠ 3 0 0 ⎜ ⎟ ⎠, D = ⎝ 0 6 ⎟ 0⎠ ⎜ ⎟ 0 ⎠, D = ⎝ 0 0 1 √1 2 √1 2 √1 3 − √13 √1 3 0 2 ⎟ ⎠, 1 3 − 23 2 3 ⎞ ⎛ 0 ⎛ ⎜ ⎜ 30. λ1 = λ2 = 0, λ3 = −2, λ4 = 2, K1 = ⎜ ⎝ 0 √1 2 0 0 ⎞ 9 2 √1 2 ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ 0 0 1 1 ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ 29. λ1 = 1, λ2 = −6, λ3 = 8, K1 = ⎝ 1 ⎠, K2 = ⎝ 0 ⎠, K3 = ⎝ 0 ⎠, P = ⎝ 1 −1 0 1 0 ⎜ ⎜ P=⎜ ⎜ ⎝ 1 ⎞ √ ⎞ √ ⎞ ⎛ ⎛ ⎞ ⎛ 1+ 2 0 1− 2 √ √ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 28. λ1 = 1, λ2 = 2 − 2 , λ3 = 2 + 2 , K1 = ⎝ 1 ⎠, K2 = ⎝ 0 ⎠, K3 = ⎝ 0 ⎠, 1 1 0 √ √ ⎞ √ √ ⎛ ⎞ ⎛ 2 2+ 2 0 − 2− 1 0 0 2 2 ⎜ ⎟ √ ⎟ ⎜ P=⎜ , D = ⎝0 2 − 2 2 0 ⎠ 0 0 ⎟ ⎝1 ⎠ √ √ √ √ √ 2+ 2 2− 2 0 0 2+ 2 0 − √12 ⎞ 0 0 5 ⎛ ⎞ ⎛ ⎞ ⎛ ⎛2 ⎞ 2 1 2 3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ 27. λ1 = 3, λ2 = 6, λ3 = 9, K1 = ⎝ 2 ⎠, K2 = ⎝ −1 ⎠, K3 = ⎝ −2 ⎠, P = ⎝ 32 1 −2 1 2 ⎛ −1 0 − √12 0 √1 2 1 2 − 12 1 2 − 12 1 ⎞ 2 1 2 ⎟ ⎟ ⎟, 1 ⎟ 2 ⎠ 1 2 −1 1 ⎛ ⎞ ⎞ 1 0 0 ⎜ ⎟ ⎟ 0 ⎠, D = ⎝ 0 −6 0 ⎠ √1 0 0 8 2 ⎛ ⎞ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0⎟ ⎜ −1 ⎟ ⎜ −1 ⎟ ⎜1⎟ ⎟, K2 = ⎜ ⎟, K3 = ⎜ ⎟, K4 = ⎜ ⎟ ⎝ 0⎠ ⎝ 1⎠ ⎝1⎠ 1⎠ ⎞ ⎛ 0 0 0 ⎜0 0 ⎜ D=⎜ ⎝0 0 0 0 ⎛ 0 0 − √12 √1 2 1 ⎞ 0 0 0 0⎟ ⎟ ⎟ −2 0 ⎠ 0 2 422 ⎞ ⎛ −1 ⎞ 1 8.12 ! T 5 −1 −1* 5 Diagonalization "! " x = 24. Using y 31. The given equation can be written as X AX = 24: ( x y ) ) 1 " ! " ! √ √1 1 1 2 2 and X = PX′ we find , P= , K2 = λ1 = 6, λ2 = 4, K1 = 1 −1 − √12 √12 ! "! " 6 0 X (X Y ) = 24 or 6X 2 + 4Y 2 = 24. Y 0 4 The conic section is an ellipse. Now from X′ = PT X we see that the XY -coordinates of (1, −1) and (1, 1) √ √ are ( 2 , 0) and (0, 2 ), respectively. From this we conclude that the X-axis and Y -axis are as shown in the accompanying figure. "! " ! x 13 −5 = 288. 32. The given equation can be written as XT AX = 288: ( x y ) y −5 13 * ) ! " ! " √1 − √12 1 −1 2 and X= PX′ we find Using λ1 = 8, λ2 = 18, K1 = , K2 = , P= 1 1 √1 √1 2 2 ! "! " 8 0 X (X Y ) = 288 or 8X 2 + 18Y 2 = 288. Y 0 18 The conic section is an ellipse. Now from X′ = PT X we see that the XY -coordinates of (1, 1) and (1, −1) are √ √ ( 2 , 0) and (0, − 2 ), respectively. From this we conclude that the X-axis and Y -axis are as shown in the accompanying figure. ! T −3 4 4* 3 "! " x = 20. Using y 33. The given equation can be written as X AX = 20: ( x y ) ) 1 " ! ! " √ − √25 −2 1 5 , P= , K2 = and X = PX′ we find λ1 = 5, λ2 = −5, K1 = 1 2 √2 √1 5 5 "! " ! X 5 0 = 20 or 5X 2 − 5Y 2 = 20. (X Y ) Y 0 −5 The conic section is a hyperbola. Now from X′ = PT X we see that the XY -coordinates of (1, 2) and (−2, 1) √ √ are ( 5 , 0) and (0, 5 ), respectively. From this we conclude that the X-axis and Y -axis are as shown in the accompanying figure. 34. The given equation can be written as XT AX = 288: ! " "! " ! x x 16 12 = 0. + ( −3 4 ) (x y) y y 12 9 * )4 ! " ! " 3 4 −3 5 5 and X = PX′ we find Using λ1 = 25, λ2 = 0, K1 = , K2 = , P= 3 4 3 4 − 5 5 ! " "! " ! X X 25 0 = 0 or 25X 2 + 5Y = 0. + (0 5) (X Y ) Y Y 0 0 The conic section is a parabola. Now from X′ = PT X we see that the XY -coordinates of (4, 3) and (3, −4) are (5, 0) and (0, −5), respectively. From this we conclude that the X-axis and Y -axis are as shown in the accompanying figure. 35. Since D = P−1 AP we have A = PDP−1 . Hence ! "! "! " ! " 1 1 2 0 −1 1 4 −1 A= = . 2 1 0 3 2 −1 2 1 423 8.12 Diagonalization 36. Since eigenvectors are mutually orthogonal we use an orthogonal matrix P and A = PDPT . ⎞ ⎛ √1 ⎞⎛ ⎞ ⎛ 8 4 ⎞ ⎛ √1 √1 √1 √1 − √13 − 13 1 0 0 3 3 3 2 6 3 3 ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ 4 ⎟ A = ⎝ − √13 0 √26 ⎠ ⎝ 0 3 0 ⎠ ⎝ √12 0 − √12 ⎠ = ⎝ 43 11 3 3 ⎠ 1 4 8 1 1 2 1 1 1 0 0 5 √ √ √ √ √ − −√ 3 2 6 6 6 3 6 3 3 37. Since D = P−1 AP we have A = PDP−1 A2 = PDP−1 PDP−1 = PDDP−1 = PD2 P−1 A3 = A2 A = PD2 P−1 PDP−1 = PD2 DP−1 = PD3 P−1 and so on. ⎛ 4 2 0 0 ⎜ 0 34 0 ⎜ 38. ⎜ ⎝ 0 0 (−1)4 ⎞ ⎛ ⎞ 0 16 0 0 0 ⎜ 0 ⎟ 0 ⎟ ⎟ ⎜ 0 81 0 ⎟ ⎟=⎜ ⎟ 0 ⎠ ⎝ 0 0 1 0 ⎠ 0 0 0 (5)4 0 0 0 625 ) 2 " ! " ! " ! 1 −1 −1 1 3 39. λ1 = 2, λ2 = −1, K1 = , P= , P−1 = , K2 = 1 2 1 2 − 31 * ) ! " " "! ! 2 1 21 11 32 0 1 −1 3 3 = A5 = 22 10 0 −1 1 2 − 31 13 " ! " ! " ! " ! 5 2 −1 2 5 2 40. λ1 = 0, λ2 = 1, K1 = , P−1 = , K2 = , P= 3 1 3 −5 3 1 ! "! "! " " ! 5 2 0 0 −1 2 6 −10 A10 = = 3 1 0 1 3 −5 3 −5 1 3 1 3 * EXERCISES 8.13 Cryptography 1. (a) The message is M = ! 19 5 14 4 0 " . The encoded message is 5 12 16 0 "! " ! ! 1 2 19 5 14 4 0 35 15 B = AM = = 1 1 8 5 12 16 0 27 10 8 38 36 0 26 20 0 " . (b) The decoded message is M=A −1 2. (a) The message is M = B = AM = ! 3 5 1 2 "! B= ! 20 0 ! −1 2 1 −1 "! 8 5 0 13 9 19 0 8 20 8 5 0 9 19 0 13 0 8 35 15 38 36 0 27 10 26 20 0 " = ! 19 5 14 4 0 8 5 12 16 0 " " 14 5 25 . The encoded message is 18 5 0 " ! 60 69 110 0 79 70 15 14 5 25 = 5 18 5 0 20 26 43 0 29 25 . 15 5 424 132 40 75 50 15 25 " . 8.13 Cryptography (b) The decoded message is M = A−1 B = ! = ! 2 −1 −5 3 "! 20 8 5 0 9 19 60 20 69 26 0 13 0 8 110 43 0 79 0 29 15 14 5 25 5 18 5 0 70 25 " 40 15 75 25 120 75 107 67 40 25 " . " 16 8 15 14 5 . The encoded message is 0 8 15 13 5 ! " ! "! 3 5 48 64 16 8 15 14 5 B = AM = = 0 8 15 13 5 2 3 32 40 3. (a) The message is M = 132 50 ! " . (b) The decoded message is M=A −1 B= ⎛ ! −3 5 2 −3 7 15 ⎜ 4. (a) The message is M = ⎝ 8 = 1 ⎜ B = AM = ⎝ 1 0 ⎛ 7 15 ⎜ 5. (a) The message is M = ⎝ 8 2 ⎜ B = AM = ⎝ 1 −1 48 32 64 40 120 75 20 0 14 15 18 15 14 0 13 (b) The decoded message is ⎞⎛ ⎛ 50 0 1 −1 ⎟⎜ ⎜ −1 M = A B = ⎝ 2 −2 −1 ⎠ ⎝ 33 −1 1 1 26 ⎛ ! 40 25 " = ⎞ 1 1 1 0 ⎞ ! 16 8 15 14 5 0 8 15 13 5 75 55 40 44 31 13 ⎞ 22 ⎟ 21 ⎠ . 1 ⎛ 0 14 15 18 20 0 15 14 0 13 9 14 0 19 20 0 ⎟ 1 ⎠. 0 99 75 44 22 43 15 66 55 31 28 15 52 40 13 ⎟ ⎜ 21 ⎠ = ⎝ 8 14 15 18 15 14 0 13 20 . 99 66 52 57 43 28 30 1 " message is 57 0 30 15 15 7 15 ⎞ ⎞ ⎟ 1 ⎠. The encoded message is 9 14 0 19 20 0 0 ⎞ ⎛ ⎞ ⎞⎛ 31 44 15 61 50 49 41 1 7 15 0 14 15 18 20 ⎟ ⎜ ⎟ ⎟⎜ 29 15 47 35 31 21 ⎠ . 1 ⎠ ⎝ 8 0 15 14 0 13 1 ⎠ = ⎝ 24 9 14 0 19 20 0 0 1 −15 15 0 −15 −5 −19 0 (b) The decoded message is ⎞⎛ ⎛ 31 1 −1 0 ⎟⎜ ⎜ M = A−1 B = ⎝ 1 −1 1 ⎠ ⎝ 24 −2 3 −1 1 ⎛ 4 18 ⎜ 6. (a) The message is M = ⎝ 14 0 8 44 29 −15 15 15 15 0 10 15 9 19 0 5 0 19 16 ⎞⎛ 5 3 0 4 18 0 ⎜ ⎟⎜ B = AM = ⎝ 4 3 −1 ⎠ ⎝ 14 0 9 8 5 0 5 2 2 ⎛ 107 67 ⎟ 1 ⎠. The encoded 9 14 0 19 20 0 0 ⎞ ⎛ ⎞⎛ 50 7 15 0 14 15 18 20 2 3 ⎟ ⎜ ⎟⎜ 1 2 ⎠ ⎝ 8 0 15 14 0 13 1 ⎠ = ⎝ 33 26 9 14 0 19 20 0 0 1 2 ⎛ 0 " 61 50 49 47 35 31 0 −15 −5 ⎞ ⎛ 41 7 15 0 ⎟ ⎜ 21 ⎠ = ⎝ 8 0 15 −19 9 14 0 ⎞ 8 ⎟ 20 ⎠. The encoded message is 14 14 19 15 18 0 13 20 0 75 44 107 ⎞ 100 ⎟ 67 ⎠ . 130 25 10 19 19 ⎞ ⎛ 62 15 8 ⎟ ⎜ 0 20 ⎠ = ⎝ 50 16 25 64 425 90 67 100 27 27 18 107 78 126 ⎞ 20 ⎟ 1 ⎠. 0 8.13 Cryptography (b) The decoded message is ⎞⎛ ⎛ 62 8 −6 −3 ⎟⎜ ⎜ −1 M = A B = ⎝ −13 10 5 ⎠ ⎝ 50 64 −7 5 3 90 67 100 27 27 18 107 78 126 ⎞ ⎛ 4 18 100 ⎟ ⎜ 67 ⎠ = ⎝ 14 0 8 5 130 75 44 107 7. The decoded message is M = A−1 B = ! 2 −5 −3 8 "! 152 95 184 171 86 212 116 107 56 133 " = ! 19 0 0 10 9 19 15 0 0 19 16 4 25 18 4 " ⎞ 8 ⎟ 20 ⎠ . 25 20 8 21 1 23 8 1 20 0 13 5 0 23 15 18 18 25 0 . From correspondence (1) we obtain: STUDY HARD. 8. The decoded message is "! ! 46 −7 −13 1 −1 −1 M=A B= 23 −15 −14 1 −2 22 −18 2 −18 1 10 −12 5 " = ! From correspondence (1) we obtain: WHAT ME WORRY . 9. The decoded message is ⎛ 0 ⎜ −1 M = A B = ⎝0 1 ⎞⎛ 1 31 ⎟⎜ 0 ⎠ ⎝ 19 −1 13 0 1 0 ⎞ ⎛ 21 21 22 20 9 13 ⎟ ⎜ 0 9 13 16 15 ⎠ = ⎝ 19 1 20 8 0 9 18 1 0 20 20 9 1 8 13 14 0 16 20 From correspondence (1) we obtain: MATH IS IMPORTANT. ⎞ 9 ⎟ 15 ⎠ . 0 10. The decoded message is ⎛ ⎞⎛ 1 0 −1 36 32 28 61 26 56 ⎜ ⎟⎜ −1 M = A B = ⎝ −1 1 2 ⎠ ⎝ −9 −2 −18 −1 −18 −25 23 27 23 41 26 43 0 −1 0 ⎞ ⎛ 13 5 5 20 0 13 5 0 ⎟ ⎜ 5 0 12 ⎠ . = ⎝ 1 20 0 20 8 9 2 18 1 18 25 0 ⎞ 10 12 ⎟ 0 0 ⎠ 5 12 0 From correspondence (1) we obtain: MEET ME AT THE LIBRARY . " ! u v −1 . Then 11. Let A = x y ! "! " u v 17 16 18 5 34 0 34 20 9 5 25 −1 A B= , x y −30 −31 −32 −10 −59 0 −54 −35 −13 −6 −50 " ! 2 1 so 17u − 30v = 4, 16u − 31v = 1 and 5x − 6y = 1, 25x − 50y = 25. Then A−1 = and −1 −1 " ! 4 1 4 0 9 0 14 5 5 4 0 . A−1 B 13 15 14 5 25 0 20 15 4 1 25 (1) we obtain: DAD I NEED MONEY TODAY. ⎞ 19 27 21 3 3 27 21 18 21 ⎟ 21 22 3 25 27 6 7 14 23 ⎠ 21 7 27 5 21 17 2 25 7 ⎞ ⎛ ⎞ 1 0 37 38 61 56 51 33 51 50 30 57 51 ⎟ ⎜ ⎟ 0 1 ⎠ = ⎝ 24 35 40 34 48 8 24 44 23 43 28 ⎠ 1 1 −1 11 −24 0 15 −24 20 6 −11 5 −11 16 From correspondence ⎛ 22 8 ⎜ T 12. (a) M = ⎝ 13 3 2 27 ⎛ 1 ⎜ T (b) B = M = ⎝ 1 426 " . 8.14 ⎛ −1 1 1 An Error-Correcting Code ⎞ ⎟ ⎜ (c) BA−1 = B ⎝ 2 −1 −1 ⎠ = M 1 0 −1 ⎞ ⎛ 15 22 20 8 23 6 21 22 ⎟ ⎜ 13. (a) B′ = ⎝ 10 22 18 23 25 2 23 25 ⎠ 3 26 26 14 23 16 26 12 (b) Using correspondence (1) the encoded message is: OVTHWFUVJVRWYBWYCZZNWPZL. ⎞ ⎛ ⎞ ⎛ 46 32 14 58 54 −34 35 86 1 4 −3 ⎟ ⎜ ⎟ ⎜ (c) M′ A−1 B′ = ⎝ 2 58 42 57 75 −14 59 95 ⎠ 3 −2 ⎠ B′ = ⎝ 54 −61 −54 −34 −66 −77 28 −56 −108 −2 −4 3 ⎞ ⎛ 19 5 14 4 0 20 8 5 ⎟ ⎜ ′ M = M mod 27 = ⎝ 0 4 15 3 21 13 5 14 ⎠. 20 0 20 15 4 1 25 0 Using correspondence (1) the encoded message is: SEND THE DOCUMENT TODAY. EXERCISES 8.14 An Error-Correcting Code 1. ( 0 1 1 0 ) 2. ( 1 1 1 1) 3. ( 0 0 0 1 1 ) 4. ( 1 0 1 0 0 ) 5. ( 1 0 1 0 1 0 0 1 ) 6. ( 0 1 1 0 1 0 1 0 ) 7. ( 1 0 0 ) 8. ( 0 0 9. Parity error 10. ( 1 0 1) 1 0) 12. Parity error 11. ( 1 0 0 1 1 ) In Problems 13-18, D = ( c1 c2 ⎛ 1 ⎜ c3 ) and P = ⎝ 1 0 T T T T T T T T T T 1 0 1 0 1 1 ⎞ 1 ⎟ 1 ⎠. 1 13. DT = P ( 1 1 1 0 ) = ( 0 0 0 ) ; C = ( 0 0 1 0 1 1 0 ) 14. DT = P ( 0 0 1 1 ) = ( 1 0 0 ) ; C = ( 1 0 0 0 0 1 1 ) 15. DT = P ( 0 1 0 1 ) = ( 0 1 0 ) ; C = ( 0 1 0 0 1 0 1 ) 16. DT = P ( 0 0 0 1 ) = ( 1 1 1 ) ; C = ( 1 1 0 1 0 0 1 ) 17. DT = P ( 0 1 1 0 ) = ( 1 1 0 ) ; C = ( 1 1 0 0 1 1 0 ) 427 8.14 An Error-Correcting Code T T 18. DT = P ( 1 1 0 0 ) = ( 0 1 1 ) ; C = ( 0 1 1 1 1 0 0 ) In Problems 19-28, W represents the correctly decoded message. T 19. S = HRT = H ( 0 0 0 0 0 0 0 ) = ( 0 0 0 ) ; a code word. W = ( 0 0 0 0 ) T not a code word. The error is in the third bit. T not a code word. The error is in the fifth bit. 20. S = HRT = H ( 1 1 0 0 0 0 0 ) = ( 0 1 1 ) ; W = (1 0 0 0) 21. S = HRT = H ( 1 1 0 1 1 0 1 ) = ( 1 0 1 ) ; W = (0 0 0 1) T a code word. W = ( 0 0 1 0 ) T a code word. W = ( 1 1 1 1 ) T a code word. W = ( 0 1 1 0 ) 22. S = HRT = H ( 0 1 0 1 0 1 0 ) = ( 0 0 0 ) ; 23. S = HRT = H ( 1 1 1 1 1 1 1 ) = ( 0 0 0 ) ; 24. S = HRT = H ( 1 1 0 0 1 1 0 ) = ( 0 0 0 ) ; T not a code word. The error is in the second bit. T not a code word. The error is in the second bit. T not a code word. The error is in the seventh bit. T not a code word. The error is in the second bit. 25. S = HRT = H ( 0 1 1 1 0 0 1 ) = ( 0 1 0 ) ; W = (1 0 0 1) 26. S = HRT = H ( 1 0 0 1 0 0 1 ) = ( 0 1 0 ) ; W = (0 0 0 1) 27. S = HRT = H ( 1 0 1 1 0 1 1 ) = ( 1 1 1 ) ; W = (1 0 1 0) 28. S = HRT = H ( 0 0 1 0 0 1 1 ) = ( 0 1 0 ) ; W = (1 0 1 1) 29. (a) 27 = 128 (b) 24 = 16 (c) ( 0 0 0 0 0 0 0 ), ( 1 1 0 1 0 0 1 ), ( 0 1 0 1 0 1 0 ), ( 1 0 0 0 0 1 1 ), ( 1 0 0 1 1 0 0 ), ( 0 1 0 0 1 0 1 ), ( 1 1 0 0 1 1 0 ), ( 0 0 0 1 1 1 1 ), ( 1 1 1 0 0 0 0 ), ( 0 0 1 1 0 0 1 ), ( 1 0 1 1 0 1 0 ), ( 0 1 1 0 0 1 1 ), ( 0 1 1 1 1 0 0 ), ( 1 0 1 0 1 0 1 ), ( 0 0 1 0 1 1 0 ), ( 1 1 1 1 1 1 1 ) 30. (a) c4 = 0, c3 ⎛ 0 ⎜0 ⎜ (b) H = ⎜ ⎝0 1 = 1, c2 = 1, c1 = 0; ( 0 1 1 0 0 1 1 0 ) ⎞ 0 0 0 1 1 1 1 0 1 1 0 0 1 1⎟ ⎟ T T ⎟ (c) S = HRT = H ( 0 0 1 1 1 1 0 0 ) = ( 0 0 0 0 ) ⎠ 1 0 1 0 1 0 1 1 1 1 1 1 1 1 428 8.15 Method of Least Squares EXERCISES 8.15 Method of Least Squares YT = ( 1 2 3 2 ) 1. We have AT A = Now T −1 so X = (A A) T A Y= )2* 5 3 5 54 14 14 4 " and Y = ( −1 3 5 7 ) T Now A A= so X = (AT A)−1 AT Y = ) * 13 5 − 25 ! 14 6 6 4 Y = ( 1 1.5 T Now A A= T −1 so X = (A A) T A Y= ! 1.1 −0.3 ! " 55 15 4 5 " . 1 1 1 ! " 4 −14 1 = 20 −14 54 1 Y = ( 0 1.5 T A A= " and T and A = T −1 (A A) ! 0 1 2 3 1 1 ! 1 1 1 = 20 " 4 −6 −6 14 . " 3 4.5 5 ) 15 5 " and T and A = T −1 (A A) ! 1 1 2 1 ! 1 = 50 3 1 4 1 5 −15 −15 55 " 5 . 1 " and the least squares line is y = 1.1x − 0.3. T Now 3 and the least squares line is y = 2.6x − 0.4. T 3. We have (AT A)−1 2 ! and the least squares line is y = 0.4x + 0.6. T 2. We have 4. We have ! and AT = ! 54 3 4.5 5 ) 14 " and T and A = T −1 (A A) ! 0 2 3 4 5 1 1 1 1 1 " 1 = 74 ! 5 −14 −14 54 " . 14 5 " 1.06757 T −1 T so X = (A A) A Y = and the least squares line is y = 1.06757x − 0.189189. −0.189189 " ! 0 1 2 3 4 5 6 T T . 5. We have Y = ( 2 3 5 5 9 8 10 ) and A = 1 1 1 1 1 1 1 ! " " ! 91 21 7 −21 1 T T −1 Now A A= and (A A) = 196 −21 21 7 91 ) 19 * ! so X = (AT A)−1 AT Y = 6. We have Now 14 27 14 YT = ( 2 2.5 AT A = and the least squares line is y = 1.35714x + 1.92857. 1 ! 1.5 2 140 28 28 7 3.2 5 ) " and 5 6 1 1 " ! 7 −28 1 = 196 −28 140 and AT = (AT A)−1 429 ! 1 1 2 1 3 1 4 1 " 7 . 1 8.15 Method of Least Squares so X = (AT A)−1 AT Y = 7. We have YT = ( 220 ! 200 T Now 0.407143 0.828571 A A= so X = (AT A)−1 AT Y = 180 ! ) " and the least squares line is y = 0.407143x + 0.828571. 170 150 36400 420 420 * 117 6 − 140 703 3 " and AT = 135 ) and (AT A)−1 = ! 20 1 40 1 ! 1 42000 60 1 6 −420 80 1 100 1 " −420 " 120 . 1 36400 and the least squares line is v = −0.835714T + 234.333. At T = 140, v ≈ 117.333 and at T = 160, v ≈ 100.619. T 8. We have Y = ( 0.47 Now T 0.90 A A= ! 2.0 3.7 7.5 15 ) 1697500 3150 T and A = " T −1 ! 400 450 500 550 600 650 1 1 ! 1 1 1 1 1 = 262500 6 −3150 " . " and (A A) 6 −3150 1697500 " ! 0.0538 and the least squares line is R = 0.0538T − 23.3167. At T = 700, so X = (AT A)−1 AT Y = −23.3167 R ≈ 14.3433. 3150 EXERCISES 8.16 Discrete Compartmental Models In Problems 1-5 we use the fact that the element τij in the transfer matrix T is the rate of transfer from compartment j to compartment i, and the fact that the sum of each column in T is 1. 1. (a) The initial state and the transfer matrix are " ! ! " 0.8 0.4 90 X0 = . and T = 0.2 0.6 60 (b) We have X1 = TX0 = ! 0.8 0.4 0.2 0.6 "! 90 X2 = TX1 = ! 0.8 0.4 0.2 0.6 "! 96 and 60 54 " = ! 96 " = ! 98.4 51.6 54 " " . (c) From TX̂ − X̂ = (T − I)X̂ = 0 and the fact that the system is closed we obtain −0.2x1 + 0.4x2 = 0 x1 + x2 = 150. The solution is x1 = 100, x2 = 50, so the equilibrium state is X̂ = 430 ! " 100 . 50 8.16 Discrete Compartmental Models 2. (a) The initial state and the transfer matrix are ⎛ ⎞ ⎛ 0.7 100 ⎜ ⎟ ⎜ X0 = ⎝ 200 ⎠ and T = ⎝ 0.3 0 150 ⎞ 0 0.5 ⎟ 0.8 0 ⎠ . 0.2 0.5 (b) We have ⎛ ⎞ 145 ⎟ ⎜ X1 = TX0 = ⎝ 190 ⎠ 115 ⎞ 159 ⎜ ⎟ and X2 = TX1 = ⎝ 195.5 ⎠ . ⎛ 95.5 (c) From TX̂ − X̂ = (T − I)X̂ = 0 and the fact that the system is closed we obtain −0.8x1 + 0.5x2 =0 0.3x1 − 0.9x2 =0 x1 + x2 + x3 = 450. ⎞ 145.161 ⎟ ⎜ The solution is x1 = 145.161, x2 = 217.742, x3 = 87.0968, so the equilibrium state is X̂ = ⎝ 217.742 ⎠. ⎛ 87.097 3. (a) The initial state and the transfer matrix are ⎛ ⎞ ⎛ 0.2 100 ⎜ ⎟ ⎜ X0 = ⎝ 0 ⎠ and T = ⎝ 0.3 0.5 0 0.5 0.1 0.4 ⎞ 0 ⎟ 0⎠. 1 (b) We have ⎛ 20 ⎞ ⎜ ⎟ X1 = TX0 = ⎝ 30 ⎠ 50 ⎛ 19 ⎞ ⎜ ⎟ and X2 = TX1 = ⎝ 9 ⎠ . 72 (c) From TX̂ − X̂ = (T − I)X̂ = 0 and the fact that the system is closed we obtain −0.8x1 + 0.5x2 =0 0.3x1 − 0.9x2 =0 x1 + x2 + x3 = 100. ⎛ 0 ⎞ ⎟ ⎜ The solution is x1 = x2 = 0, x3 = 100, so the equilibrium state is X̂ = ⎝ 0 ⎠. 100 4. (a) The transfer matrix is ⎛ 0.7 0.05 ⎜ T = ⎝ 0.3 0.75 0 0.2 431 ⎞ 0.15 ⎟ 0 ⎠. 0.85 8.16 Discrete Compartmental Models (b) Year Bare Space Grasses Small Shrubs 0 10.00 0.00 0.00 1 2 7.00 5.05 3.00 4.35 0.00 0.60 3 3.84 4.78 1.38 4 5 3.14 2.75 4.74 4.49 2.13 2.76 6 2.56 4.19 3.24 5. From TX̂ = 1X̂ we see that the equilibrium state vector X̂ is the eigenvector of the transfer matrix T corresponding to the eigenvalue 1. It has the properties that its components add up to the sum of the components of the initial state vector. 6. (a) The initial state and the transfer matrix are ⎛ ⎞ ⎛ 0.88 0 ⎜ ⎟ ⎜ X0 = ⎝ 100 ⎠ and T = ⎝ 0.06 0.06 0 (b) Year Phytoplankton Water Zooplankton 0 1 0.00 2.00 100.00 97.00 0.00 1.00 2 3.70 94.26 2.04 3 4 5.14 6.36 91.76 89.47 3.10 4.17 5 6 7.39 8.25 87.37 85.46 5.24 6.30 7 8.97 83.70 7.33 8 9 9.56 10.06 82.10 80.62 8.34 9.32 10 11 10.46 10.79 79.28 78.04 10.26 11.17 12 11.06 76.90 12.04 ⎞ 0 ⎟ 0.05 ⎠ . 0.02 0.97 0.01 0.95 CHAPTER 8 REVIEW EXERCISES 2 ⎜3 ⎜ 1. ⎜ ⎝4 5 ⎛ 3 4 5 6 3. AB = ! ⎞ 4 5⎟ ⎟ ⎟ 6⎠ 2. 4 × 3 3 6 4. A−1 = − 7 " 4 ; BA = ( 11 ) 8 432 1 2 ! 4 −3 −2 1 " = ! −2 3 2 1 − 12 " CHAPTER 8 REVIEW EXERCISES 5. False; consider A = 7. det ! 1 0 0 1 + 1 , + 1 ,3 (5) = 2A = 2 5 8 " and B = ! 0 1 1 0 " 6. True ; det(−AT ) = (−1)3 (5) = −5 8. det AB−1 = det A/ det B = 6/2 = 3 9. 0 10. det C = (−1)3 / det B = −1/103 (2) = −1/2000 12. True 11. False; an eigenvalue can be 0. 13. True 14. True, since complex roots of real polynomials occur in conjugate pairs. 15. False; if the characteristic equation of an n×n matrix has repeated roots, there may not be n linearly independent eigenvectors. 16. True 17. True 18. True 19. False; A is singular and thus not orthogonal. 20. True 21. A = 21 (A + AT ) + 21 (A − AT ) where 21 (A + AT ) is symmetric and 12 (A − AT ) is skew-symmetric. ! " ! " 0 1 0 1 22. Since det A2 = (det A)2 ≥ 0 and det = −1, there is no A such that A2 = . 1 0 1 0 " ! 1 1 is nilpotent. 23. (a) −1 −1 (b) Since det An = (det A)n = 0 we see that det A = 0 and A is singular. " " ! " ! ! 0 i 0 −1 i 0 24. (a) σx σy = = −σz σy = −σz σx ; σy σz = = −σy σx ; σx σz = i 0 1 0 0 −i (b) We first note that for anticommuting matrices AB = −BA, so C = 2AB. Then Cxy " " ! ! 0 2 0 2i Cyz = . , and Czx = −2 0 2i 0 ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ ⎛ 9 5 −1 1 −9 1 1 5 1 1 5 9 1 R13 row row ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ 4 0 27 ⎠ −−−−− 4 0 27 ⎠ −−−−−−→ ⎝ 0 1 −5 92 ⎠ −−−−−−→ ⎝ 0 −→ ⎝ 2 25. ⎝ 2 operations operations 0 0 1 12 1 1 5 5 −1 1 −9 0 9 The solution is ⎛ 1 1 1 ⎜ 26. ⎝ 1 −2 3 2 0 −3 T X = ( − 21 7 12 ) . ⎞ ⎛ 6 1 1 row ⎟ ⎜ 2 ⎠ −−−−−−→ ⎝ 0 1 operations 0 0 3 The solution is x1 = 3, x2 = 2, x3 = 1. 1 − 23 1 6 ⎞ ⎛ 1 0 0 row ⎜ 4 ⎟ 3 ⎠ −−−−−−→ ⎝ 0 1 0 operations 0 0 1 1 3 = ! " 2i 0 , 0 −2i 0 0 1 0 0 1 − 12 ⎞ ⎟ 7 ⎠. 1 2 ⎞ ⎟ 2 ⎠. 1 27. Multiplying the second row by abc we obtain the third row. Thus the determinant is 0. 28. Expanding along the first row we see that the result is an expression of the form ay + bx2 + cx + d = 0, which is a parabola since, in this case a ̸= 0 and b ̸= 0. Letting x = 1 and y = 2 we note that the first and second rows are the same. Similarly, when x = 2 and y = 3, the first and third rows are the same; and when x = 3 433 CHAPTER 8 REVIEW EXERCISES and y = 5, the first and fourth rows are the same. In each case the determinant is 0 and the points lie on the parabola. 29. 4(−2)(3)(−1)(2)(5) = 240 30. (−3)(6)(9)(1) = −162 ⎞ ⎛ 1 −1 1 ⎟ ⎜ 1 −1 ⎠ = 18 = 0, the system has only the trivial solution. 31. Since ⎝ 5 1 2 1 ⎞ ⎛ 1 −1 −1 ⎟ ⎜ 1 −1 ⎠ = 0, the system has infinitely many solutions. 32. Since ⎝ 5 1 2 1 33. From x1 I2 + x2 HNO3 → x3 HIO3 + x4 NO2 + x5 H2 O we obtain the system 2x1 = x3 , x2 = x3 + 2x5 , x2 = x4 , 3x2 = 3x3 +2x4 +x5 . Letting x4 = x2 in the fourth equation we obtain x2 = 3x3 +x5 . Taking x1 = t we see that x3 = 2t, x2 = 2t + 2x5 , and x2 = 6t + x5 . From the latter two equations we get x5 = 4t. Taking t = 1 we have x1 = 1, x2 = 10, x3 = 2, x4 = 10, and x5 = 4. The balanced equation is I2 +10HNO3 → 2HIO3 +10NO2 +4H2 O. 34. From x1 Ca + x2 H3 PO4 → x3 Ca3 P2 O8 + x4 H2 we obtain the system x1 = 3x3 , 3x2 = 2x4 , x2 = 2x3 , 4x2 = 8x3 . Letting x3 = t we see that x1 = 3t, x2 = 2t, and x4 = 3t. Taking t = 1 we obtain the balanced equation 3Ca + 2H3 PO4 → Ca3 P2 O8 + 3H2 . 42 1 −21 1 −56 2 = − , x2 = = , x3 = = −84 2 −84 4 −84 3 −4 0 16 = 4, x2 = = −1, x3 = = 0 36. det = 4, det A1 = 16, det A2 = −4, det A3 = 0; x1 = 4 4 4 35. det A = −84, det A1 = 42, det A2 = −21, det A3 = −56; x1 = 37. det A = cos2 θ + sin2 θ, det A1 = X cos θ − Y sin θ, det A2 = Y cos θ + X sin θ; x1 = X cos θ − Y sin θ, y = Y cos θ + X sin θ 38. (a) i1 − i2 − i3 − i4 = 0, i2 R1 = E, i2 R1 − i3 R2 = 0, i3 R2 − i4 R3 = 0 ⎛ ⎞ 1 −1 −1 −1 ⎜0 R 0 0⎟ 1 ⎜ ⎟ (b) det A = ⎜ ⎟ = R1 R2 R3 ; ⎝ 0 R1 −R2 0⎠ 0 0 R2 −R3 ⎞ ⎛ 0 −1 −1 −1 ⎜E R 0 0⎟ 1 ⎟ ⎜ det A1 = ⎜ ⎟ = −E[−R2 R3 − R1 (R3 + R2 )] = E(R2 R3 + R1 R3 + R1 R2 ); ⎝ 0 R1 −R2 0⎠ 0 0 R2 −R3 ! " det A1 1 1 1 E(R2 R3 + R1 R3 + R1 R2 ) =E + + = det A R1 R2 R 3 R1 R2 R3 ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎛ ⎞ −2 −3 −2 x1 2 3 −1 6 1⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ 0 −1 ⎠, we have 0 ⎠ ⎝ x2 ⎠ = ⎝ −3 ⎠. Since A−1 = − ⎝ −1 39. AX = B is ⎝ 1 −2 3 −4 −6 −7 −2 0 1 9 x3 ⎛ ⎞ 7 ⎜ ⎟ X = A−1 B = ⎝ 5 ⎠. 23 i1 = 434 CHAPTER 8 REVIEW EXERCISES ⎛ 3 2 ⎜ 40. (a) A−1 B = ⎝ −1 1 2 ⎛ 3 2 ⎜ (b) A−1 B = ⎝ −1 1 2 − 14 − 94 − 14 1 2 − 14 − 94 3 2 − 14 1 2 − 14 ⎞⎛ ⎞ ⎛ ⎞ 1 −1 ⎜ ⎟ 3 ⎟⎜ ⎟ 2 ⎠⎝1⎠ = ⎝ 1⎠ 1 0 −1 4 ⎞ ⎞ ⎛ −10 −2 ⎟ ⎟ ⎜ ⎟⎜ ⎠⎝ 1⎠ = ⎝ 7⎠ −2 3 ⎞⎛ 41. From the characteristic equation λ2 − 4λ − 5 = 0 we!see that " the eigenvalues are λ1 = −1 and λ2 = 5. For −1 λ1 = −1 we have 2k1 +2k2 = 0, 4k1 4k2 = 0 and K1 = . For λ2 = 5 we have −4k1 +2k2 = 0, 4k1 −2k2 = 0 1 ! " 1 and K2 = . 2 2 42. From the characteristic ! "equation λ = 0 we see that the eigenvalues are λ1 = λ2 = 0. For λ1 = λ2 = 0 we have 0 4k1 = 0 and K1 = is a single eigenvector. 1 43. From the characteristic equation −λ3 + 6λ2 + 15λ + 8 = −(λ + 1)2 (λ − 8) = 0 we see that the eigenvalues are λ1 = λ2 = −1 and λ3 = 8. For λ1 = λ2 = −1 we have ⎛ ⎞ ⎞ ⎛ 4 2 4 0 1 21 1 0 row ⎜ ⎟ ⎟ ⎜ ⎝ 2 1 2 0 ⎠ −−−−−−→ ⎝ 0 0 0 0 ⎠ . operations 4 2 4 0 0 0 0 0 Thus K1 = ( 1 −2 T T 0 ) and K2 = ( 1 0 −1 ) . For λ3 = 8 we have ⎞ ⎛ ⎛ 1 − 25 − 54 −5 2 4 0 row ⎟ ⎜ ⎜ 2 0 ⎠ −−−−−−→ ⎝ 0 1 − 21 ⎝ 2 −8 operations 4 2 −5 0 0 0 0 0 ⎞ ⎟ 0⎠. 0 T Thus K3 = ( 2 1 2 ) . 44. From the characteristic equation −λ3 +18λ2 −99λ+162 = −(λ−9)(λ−6)(λ−3) = 0 we see that the eigenvalues are λ1 = 9, λ2 = 6, and λ3 = 3. For λ1 = 9 we have ⎛ ⎞ ⎛ −2 −2 0 0 1 1 0 row ⎜ ⎟ ⎜ 2 0 ⎠ −−−−−−→ ⎝ 0 1 −2 ⎝ −2 −3 operations 0 2 −4 0 0 0 0 Thus K1 = ( −2 2 T 1 ) . For λ2 = 6 we have ⎛ ⎞ ⎛ 1 −2 0 0 1 −2 row ⎜ ⎟ ⎜ 0 2 0 ⎠ −−−−−−→ ⎝ 0 1 ⎝ −2 operations 0 2 −1 0 0 0 0 ⎟ 0⎠. 0 ⎞ 0 ⎟ 0⎠. 0 − 21 0 0 T Thus K2 = ( 2 1 2 ) . For λ3 = 3 we have ⎞ ⎛ ⎛ 1 − 12 0 4 −2 0 0 row ⎟ ⎜ ⎜ 3 2 0 ⎠ −−−−−−→ ⎝ 0 1 1 ⎝ −2 operations 0 2 2 0 0 0 0 T Thus K3 = ( 1 2 −2 ) . 435 ⎞ 0 ⎞ ⎟ 0⎠. 0 CHAPTER 8 REVIEW EXERCISES 45. From the characteristic equation −λ3 − λ2 + 21λ + 45 = −(λ + 3)2 (λ − 5) = 0 we see that the eigenvalues are λ1 = λ2 = −3 and λ3 = 5. For λ1 = λ2 = −3 we have ⎛ ⎞ ⎛ 1 2 −3 0 1 2 −3 row ⎜ ⎟ ⎜ 4 −6 0 ⎠ −−−−−−→ ⎝ 0 0 0 ⎝ 2 operations −1 −2 3 0 0 0 0 Thus K1 = ( −2 T 1 0 ⎟ 0⎠. 0 T 0 ) and K2 = ( 3 0 1 ) . For λ3 = 5 we have ⎞ ⎛ ⎛ 1 − 72 −7 2 −3 0 row ⎟ ⎜ ⎜ 1 ⎝ 2 −4 −6 0 ⎠ −−−−−−→ ⎝ 0 operations −1 −2 −5 0 0 0 3 7 0 2 ⎟ 0⎠. ⎞ 0 0 T Thus K3 = ( −1 −2 ⎞ 1) . 46. From the characteristic equation −λ3 + λ2 + 2λ = −λ(λ + 1)(λ − 2) = 0 we see that the eigenvalues are λ1 = 0, λ2 = −1, and λ3 = 2. For λ1 = 0 we have have ⎛ 1 0 ⎜ ⎝0 1 2 2 k3 = 0, 2k1 + 2k2 + k3 = 0 and K1 = ( 1 −1 0 1 2 T ⎞ ⎛ 1 0 0 0 row ⎟ ⎜ 0 ⎠ −−−−−−→ ⎝ 0 1 1 operations 0 0 0 0 T 0 ) . For λ2 = −1 we ⎞ 0 ⎟ 0⎠. 0 Thus K2 = ( 0 1 −1 ) . For λ3 = 2 we have ⎛ ⎞ ⎛ −2 0 0 0 1 0 0 row ⎜ ⎟ ⎜ 1 0 0 −2 1 0 1 − − − − − − −→ ⎝ ⎠ ⎝ 2 operations 2 2 −1 0 0 0 0 0 ⎞ ⎟ 0⎠. 0 T Thus K3 = ( 0 1 2 ) . T 47. Let X1 = ( a XT1 ( √13 b c) √1 3 √1 3 T be the first column of the matrix. Then XT1 ( − √12 ) = √1 (a 3 T √1 ) 2 0 = √1 (c 2 − a) = 0 and + b + c) = 0. Also XT1 X1 = a2 + b2 + c2 = 1. We see that c = a and b = −2a from the first two equations. Then a2 + 4a2 + a2 = 6a2 = 1 and a = √1 6 . Thus X1 = ( √16 − √26 T √1 ) . 6 T 48. (a) Eigenvalues are λ1 = λ2 = 0 and λ3 = 5 with corresponding eigenvectors K1 = ( 0 1 0 ) , K2 = √ √ T T ( 2 0 1 ) , and K3 = ( −1 0 2 ) . Since ∥K1 ∥ = 1, ∥K2 ∥ = 5 , and ∥K3 ∥ = 5 , we have ⎛ ⎞ ⎛ ⎞ 0 1 0 0 √25 − √15 ⎜ ⎟ ⎜ 2 1 ⎟ −1 T P = ⎝1 0 0 ⎠ and P = P = ⎝ √5 0 √5 ⎠ . 0 ⎛ 0 0 ⎜ (b) P−1 AP = ⎝ 0 0 ⎟ 0⎠ 49. We identify A = (X Y )D ! X Y " 0 1 3 2 5 3 " 2 1 √2 5 − √15 0 √2 5 ⎞ 0 0 ! √1 5 . Eigenvalues are λ1 = − 21 and λ2 = 5 2 so D = = − 12 X 2 + 52 Y 2 = 1. The graph is a hyperbola. 436 ! − 12 0 0 5 2 " and the equation becomes CHAPTER 8 REVIEW EXERCISES 50. We measure years in units of 10, with 0 corresponding to 1890. Then Y = ( 63 76 ! " ! "T 30 10 0 1 2 3 4 A= , so AT A = . Thus 10 5 1 1 1 1 1 " ! " ! 15 5 −10 1 T −1 T T X = (A A) A Y = A Y= , 50 −10 30 62 92 106 T 123 ) and and the least squares line is y = 15t + 62. At t = 5 (corresponding to 1940) we have y = 137. The error in the predicted population is 5 million or 3.7%. 51. The encoded message is B = AM = ! 10 1 9 1 "! 19 14 1 20 5 12 3 8 5 4 = ! 204 185 13 12 208 188 55 50 124 112 "! 19 18 5 22 72 67 49 46 0 30 0 29 12 9 0 15 120 108 105 96 20 5 0 12 1 21 14 0 6 18 9 0 214 194 50 45 6 138 6 126 19 18 " 210 189 " . 52. The encoded message is B = AM = ! 10 1 9 1 = ! 208 189 3 0 1 19 0 20 91 84 7 21 145 131 14 20 0 1 18 5 19 0 1 13 " 219 0 11 193 . 199 0 10 175 " 53. The decoded message is ⎛ ⎞⎛ ⎞ ⎞ ⎛ −3 2 −1 19 0 15 14 0 20 8 5 12 16 0 9 ⎜ ⎟⎜ ⎟ ⎟ ⎜ M = A−1 B = ⎝ 1 0 0 ⎠ ⎝ 35 10 27 53 1 54 ⎠ = ⎝ 19 0 15 14 0 20 ⎠ . 2 −1 1 8 5 0 23 1 25 5 15 −3 48 2 39 From correspondence (1) we obtain: HELP IS ON THE WAY. 54. The decoded message is ⎛ −3 ⎜ −1 M=A B=⎝ 1 2 ⎞⎛ 2 −1 5 ⎟⎜ 0 0 ⎠ ⎝ 27 −1 1 21 From correspondence (1) we obtain: ROSEBUD . 55. (a) The parity is even so the decoded message is ( 1 1 0 ⎞ ⎛ 2 21 18 ⎟ ⎜ 17 40 ⎠ = ⎝ 5 13 −2 4 0 1) (b) The parity is odd; there is a parity error. ⎛ ⎞ ⎛ ⎞⎛ ⎞ 1 ⎛ ⎞ 1 1 0 1 ⎜ ⎟ c1 0 ⎜ ⎟⎜ ⎟⎜0⎟ ⎜ ⎟ 56. From ⎝ c2 ⎠ ⎝ 1 0 1 1 ⎠ ⎜ ⎟ = ⎝ 0 ⎠ we obtain the codeword ( 0 ⎝0⎠ c3 0 1 1 1 1 1 437 ⎞ 15 19 ⎟ 2 21 ⎠ . 0 0 0 1 1 0 0 1 ). 9 Vector Calculus EXERCISES 9.1 Vector Functions 1. 2. 3. 4. 5. 6. 7. 8. 9. Note: the scale is distorted in this graph. For t = 0, the graph starts at (1, 0, 1). The upper loop shown intersects the xz-plane at about (286751, 0, 286751). 438 9.1 10. 11. x = t, y = t, z = t2 + t2 = 2t2 ; r(t) = ti + tj + 2t2 k √ √ √ 12. x = t, y = 2t, z = ± t2 + 4t2 + 1 = ± 5t2 − 1 ; r(t) = ti + 2tj ± 5t2 − 1 k 13. x = 3 cos t, z = 9 − 9 cos2 t = 9 sin2 t, y = 3 sin t; r(t) = 3 cos ti + 3 sin tj + 9 sin2 tk 14. x = sin t, z = 1, y = cos t; r(t) = sin ti + cos tj + k 439 Vector Functions 9.1 Vector Functions 15. r(t) = sin 2t ln t i + (t − 2)5 j + k. Using L’Hôpital’s Rule, t 1/t " ! 1/t 2 cos 2t 5 i + (t − 2) j + k = 2i − 32j. lim r(t) = 1 −1/t2 t→0+ 16. (a) limt→α [−4r1 (t) + 3r2 (t)] = −4(i − 2j + k) + 3(2i + 5j + 7k) = 2i + 23j + 17k (b) limt→α r1 (t) · r2 (t) = (i − 2j + k) · (2i + 5j + 7k) = −1 17. r′ (t) = 1 2 1 1 i − 2 j; r′′ (t) = − 2 i + 3 j t t t t 18. r′ (t) = ⟨−t sin t, 1 − sin t⟩; r′′ (t) = ⟨−t cos t − sin t, − cos t⟩ 19. r′ (t) = ⟨2te2t + e2t , 3t2 , 8t − 1⟩; r′′ (t) = ⟨4te2t + 4e2t , 6t, 8⟩ 1 2t 20. r′ (t) = 2ti + 3t2 j + k; r′′ (t) = 2i + 6tj − k 1 + t2 (1 + t2 )2 21. r′ (t) = −2 sin ti + 6 cos tj √ r′ (π/6) = −i + 3 3 j 23. r′ (t) = j − 8t k (1 + t2 )2 r′ (1) = j − 2k 22. r′ (t) = 3t2 i + 2tj r′ (−1) = 3i − 2j 24. r′ (t) = −3 sin ti + 3 cos tj + 2k √ √ −3 2 3 2 ′ r (π/4) = i+ j + 2k 2 2 1 1 8 25. r(t) = ti + t2 j + t3 k; r(2) = 2i + 2j + k; r′ (t) = i + tj + t2 k; r′ (2) = i + 2j + 4k 2 3 3 Using the point (2, 2, 8/3) and the direction vector r′ (2), we have x = 2 + t, y = 2 + 2t, z = 8/3 + 4t. 6t 6 3 26. r(t) = (t3 −t)i+ j+(2t+1)2 k; r(1) = 3j+9k; r′ (t) = (3t2 −1)i+ j+(8t+4)k; r′ (1) = 2i+ j+12k. t+1 (t + 1)2 2 Using the point (0, 3, 9) and the direction vector r′ (1), we have x = 2t, y = 3 + 23 t, z = 9 + 12t. d [r(t) × r′ (t)] = r(t) × r′′ (t) + r′ (t) × r′ (t) = r(t) × r′′ (t) dt d d [r(t) · (tr(t))] = r(t) · (tr(t)) + r′ (t) · (tr(t)) = r(t) · (tr′ (t) + r(t)) + r′ (t) · (tr(t)) 28. dt dt = r(t) · (tr′ (t)) + r(t) · r(t) + r′ (t) · (tr(t)) = 2t(r(t) · r′ (t)) + r(t) · r(t) 27. 440 9.1 29. Vector Functions d d [r(t) · (r′ (t) × r′′ (t))] = r(t) · (r′ (t) × r′′ (t)) + r′ (t) · (r′ (t) × r′′ (t)) dt dt = r(t) · (r′ (t) × r′′′ (t) + r′′ (t) × r′′ (t)) + r′ (t) · (r′ (t) × r′′ (t)) = r(t) · (r′ (t) × r′′′ (t)) 30. d d [r1 (t) × (r2 (t) × r3 (t))] = r1 (t) × (r2 (t) × r3 (t)) + r′1 (t) × (r2 (t) × r3 (t)) dt dt = r1 (t) × (r2 (t) × r′3 (t) + r′2 (t) × r3 (t)) + r′1 (t) × (r2 (t) × r3 (t)) = r1 (t) × (r2 (t) × r′3 (t)) + r1 (t) × (r′2 (t) × r3 (t)) + r1 (t) × (r2 (t) × r3 (t)) # %& $ d 1 $1% 1 31. r1 (2t) + r2 = 2r′1 (2t) − 2 r′2 dt t t t d 3 2 [t r(t )] = t3 (2t)r′ (t2 ) + 3t2 r(t2 ) = 2t4 r′ (t2 ) + 3t2 r(t2 ) dt " " !' 2 " !' 2 !' 2 ' 2 (2 (2 1 ((2 3 ( ( 4t3 dt k = t2 ( i + t3 ( j + t4 ( k = i + 9j + 15k 3t2 dt j + t dt i + r(t) dt = 33. 2 2 −1 −1 −1 −1 −1 −1 −1 " " !' 4 " !' 4 !' 4 ' 4 √ √ − t dt j + sin πt dt k 2t + 1 dt i + r(t) dt = 34. 32. 0 0 0 0 (4 (4 (4 ( ( ( 26 1 16 2 1 i− j = (2t + 1)3/2 (( i − t3/2 (( j − cos πt (( k = 3 3 π 3 3 0 0 0 !' " !' " !' " ' 2 35. r(t) dt = tet dt i + −e−2t dt j + tet dt k = [tet − et + c1 ]i + #1 & #1 2 & 1 2 1 e−2t + c2 j + et + c3 k = et (t − 1)i + e−2t j + et k + c, 2 2 2 2 where c = c1 i + c2 j + c3 k. " !' " !' " !' ' 1 t t2 dt i + dt j + dt k 36. r(t) dt = 1 + t2 1 + t2 1 + t2 !' ) * " #1 & 1 = [tan−1 t + c1 ]i + ln(1 + t2 ) + c2 j + 1− dt k 2 1 + t2 & #1 ln(1 + t2 ) + c2 j + [t − tan−1 t + c3 ]k = [tan−1 t + c1 ]i + 2 1 −1 = tan ti + ln(1 + t2 )j + (t − tan−1 t)k + c, 2 where c = c1 i + c2 j + c3 k. !' " !' " !' " ' 37. r(t) = r′ (t) dt = 6 dt i + 6t dt j + 3t2 dt k = [6t + c1 ]i + [3t2 + c2 ]j + [t3 + c3 ]k Since r(0) = i − 2j + k = c1 i + c2 j + c3 k, c1 = 1, c2 = −2, and c3 = 1. Thus, r(t) = (6t + 1)i + (3t2 − 2)j + (t3 + 1)k. !' " !' " ' 1 1 38. r(t) = r′ (t) dt = t sin t2 dt i + − cos 2t dt j = −[ cos t2 + c1 ]i + [− sin 2t + c2 ]j 2 2 Since r(0) = 23 i = (− 12 + c1 )i + c2 j, c1 = 2 and c2 = 0. Thus, % $ 1 1 r(t) = − cos t2 + 2 i − sin 2tj. 2 2 !' " !' " !' " ' 39. r′ (t) = r′′ (t) dt = 12t dt i + −3t−1/2 dt j + 2 dt k = [6t2 + c1 ]i + [−6t1/2 + c2 ]j + [2t + c3 ]k 441 9.1 Vector Functions Since r′ (1) = j = (6 + c1 )i + (−6 + c2 )j + (2 + c3 )k, c1 = −6, c2 = 7, and c3 = −2. Thus, r′ (t) = (6t2 − 6)i + (−6t1/2 + 7)j + (2t − 2)k. !' " !' " !' " ' r(t) = r′ (t) dt = (6t2 − 6) dt i + (−6t1/2 + 7) dt j + (2t − 2) dt k = [2t3 − 6t + c4 ]i + [−4t3/2 + 7t + c5 ]j + [t2 − 2t + c6 ]k. Since r(1) = 2i − k = (−4 + c4 )i + (3 + c5 )j + (−1 + c6 )k, c4 = 6, c5 = −3, and c6 = 0. Thus, r(t) = (2t3 − 6t + 6)i + (−4t3/2 + 7t − 3)j + (t2 − 2t)k. " " !' " !' !' ' − sin t dt k cos t dt j + sec2 t dt i + 40. r′ (t) = r′′ (t) dt = = [tan t + c1 ]i + [sin t + c2 ]j + [cos t + c3 ]k Since r′ (0) = i + j + k = c1 i + c2 j + (1 + c3 )k, c1 = 1, c2 = 1, and c3 = 0. Thus, r′ (t) = (tan t + 1)i + (sin t + 1)j + cos tk. !' " !' " !' " ' ′ r(t) = r (t) dt = (tan t + 1) dt i + (sin t + 1) dt j + cos t dt k = [ln | sec t| + t + c4 ]i + [− cos t + t + c5 ]j + [sin t + c6 ]k. Since r(0) = −j + 5k = c4 i + (−1 + c5 )j + c6 k, c4 = 0, c5 = 0, and c6 = 5. Thus, r(t) = (ln | sec t| + t)i + (− cos t + t)j + (sin t + 5)k. + √ 41. r′ (t) = −a sin ti + a cos tj + ck; ∥r′ (t)∥ = (−a sin t)2 + (a cos t)2 + c2 = a2 + c2 ' 2π + (2π + + ( s= a2 + c2 dt = a2 + c2 t ( = 2π a2 + c2 0 0 42. r′ (t) = i + (cos t − t sin t)j + (sin t + t cos t)k + √ ∥r′ (t)∥ = 12 + (cos t − t sin t)2 + (sin t + t cos t)2 = 2 + t2 ) + ' π+ (* ((π π + ( + + √ t ( ( s= 2 + t2 dt = 2 + t2 + ln (t + 2 + t2 ( (( = 2 + π 2 + ln(π + 2 + π 2 ) − ln 2 2 2 0 0 43. r′ (t) = (−2et sin 2t + et cos 2t)i + (2et cos 2t + et sin 2t)j + et k + √ √ ∥r′ (t)∥ = 5e2t cos2 2t + 5e2t sin2 2t + e2t = 6e2t = 6 et ' 3π √ √ ((3π √ 6 et dt = 6 et ( = 6 (e3π − 1) s= 0 0 , √ √ √ 44. r′ (t) = 3i + 2 3 tj + 2t2 k; ∥r′ (t)∥ = 32 + (2 3 t)2 + (2t2 )2 = 9 + 12t2 + 4t4 = 3 + 2t2 (1 ' 1 ( 11 2 2 (3 + 2t2 ) dt = (3t + t3 ) (( = 3 + = s= 3 3 3 0 0 ' t + 2 ′ ′ 2 2 2 45. r (t) = −a sin ti + a cos tj; ∥r (t)∥ = a sin t + a cos t = a, a > 0; s = a du = at 0 r(s) = a cos(s/a)i + a sin(s/a)j; r′ (s) = − sin(s/a)i + cos(s/a)j , ∥r′ (s)∥ = sin2 (s/a) + cos2 (s/a) = 1 442 9.2 Motion on a Curve √ √ 2 2 1 46. r′ (s) = − √ sin(s/ 5 )i + √ cos(s/ 5 )j + √ k 5 5 5 √ √ 4 4 1 4 1 2 ′ 2 ∥r (s)∥ = sin (s/ 5 ) + cos (s/ 5 ) + = + =1 5 5 5 5 5 d d d 2 d (r · r) = ∥r∥2 = c = 0 and (r · r) = r · r′ + r′ · r = 2r · r′ , we have r · r′ = 0. Thus, r′ is dt dt dt dt perpendicular to r. 47. Since 48. Since ∥r(t)∥ is the length of r(t), ∥r(t)∥ = c represents a curve lying on a sphere of radius c centered at the origin. 49. Let r1 (t) = x(t)i + y(t)j. Then d d [u(t)r1 (t)] = [u(t)x(t)i + u(t)y(t)j] = [u(t)x′ (t) + u′ (t)x(t)]i + [u(t)y ′ (t) + u′ (t)y(t)]j dt dt = u(t)[x′ (t)i + y ′ (t)j] + u′ (t)[x(t)i + y(t)j] = u(t)r′1 (t) + u′ (t)r1 (t). 50. Let r1 (t) = x1 (t)i + y1 (t)j and r2 (t) = x2 (t)i + y2 (t)j. Then d d [r1 (t) · r2 (t)] = [x1 (t)x2 (t) + y1 (t)y2 (t)] = x1 (t)x′2 (t) + x′1 (t)x2 (t) + y1 (t)y2′ (t) + y1′ (t)y2 (t) dt dt = [x1 (t)x′2 (t) + y1 (t)y2′ (t)] + [x′1 (t)x2 (t) + y1′ (t)y2 (t)] = r1 (t) · r′2 (t) + r′1 (t) · r2 (t). 51. d r1 (t + h) × r2 (t + h) − r1 (t) × r2 (t) [r1 (t) × r2 (t)] = lim h→0 dt h r1 (t + h) × r2 (t + h) − r1 (t + h) × r2 (t) + r1 (t + h) × r2 (t) − r1 (t) × r2 (t) = lim h→0 h r1 (t + h) × [r2 (t + h) − r2 (t)] [r1 (t + h) − r1 (t)] × r2 (t) = lim + lim h→0 h→0 h h ) * ) * r2 (t + h) − r2 (t) r1 (t + h) − r1 (t) = r1 (t) × lim + lim × r2 (t) h→0 h→0 h h = r1 (t) × r′2 (t) + r′1 (t) × r2 (t) 52. Let v = ai + bj and r(t) = x(t)i + y(t)j. Then ' ' b ' b [ax(t) + by(t)] dt = a v · r(t) dt = a b x(t) dt + b a a EXERCISES 9.2 Motion on a Curve 1. v(t) = 2ti + t3 j; v(1) = 2i + j; ∥v(1)∥ = √ 4+1= √ 5; 2 a(t) = 2i + 3t j; a(1) = 2i + 3j 443 ' a b y(t) dt = v · ' a b r(t) dt. 9.2 Motion on a Curve √ √ 2 j; v(1) = 2i − 2j; ∥v(1)∥ = 4 + 4 = 2 2 ; 3 t 6 a(t) = 2i + 4 j; a(1) = 2i + 6j t 2. v(t) = 2ti − 3. v(t) = −2 sinh 2ti+2 cosh 2tj; v(0) = 2j; ∥v(0)∥ = 2; a(t) = −4 cosh 2ti + 4 sinh 2tj; a(0) = −4i √ + √ 1 4. v(t) = −2 sin ti + cos tj; v(π/3) = − 3 i + j; ∥v(π/3)∥ = 3 + 1/4 = 13/2; 2 √ 3 a(t) = −2 cos ti − sin tj; a(π/3) = −i − j 2 √ √ 5. v(t) = (2t − 2)j + k; v(2) = 2j + k ∥v(2)∥ = 4 + 1 = 5 ; a(t) = 2j; a(2) = 2j 6. v(t) = i+j+3t2 k; v(2) = i+j+12k; ∥v(2)∥ = a(2) = 12k √ 1 + 1 + 144 = 7. v(t) = i + 2tj + 3t2 k; v(1) = i + 2j + 3k; ∥v(1)∥ = √ √ 146 ; a(t) = 6tk; 1+4+9= √ 14 ; a(t) = 2j + 6tk; a(1) = 2j + 6k 8. v(t) = i + 3t2 j + k; v(1) = i + 3j + k; ∥v(1)∥ = a(t) = 6tj; a(1) = 6j √ 1+9+1= √ 11 ; 9. The particle passes through the xy-plane when z(t) = t2 − 5t = 0 or t = 0, 5 which gives us the points (0, 0, 0) and (25, 115, 0). v(t) = 2ti + (3t2 − 2)j + (2t − 5)k; v(0) = −2j − 5k, v(5) = 10i + 73j + 5k; a(t) = 2i + 6tj + 2k; a(0) = 2i + 2k, a(5) = 2i + 30j + 2k 10. If a(t) = 0, then v(t) = c1 and r(t) = c1 t + c2 . The graph of this equation is a straight line. 444 9.2 Motion on a Curve √ 11. Initially we are given s0 = 0 and v0 = (480 cos 30◦ )i + (480 sin 30◦ )j = 240 3 i + 240j. Using a(t) = −32j we find ' v(t) = a(t) dt = −32tj + c √ 240 3 i + 240j = v(0) = c √ √ v(t) = −32tj + 240 3 i + 240j = 240 3 i + (240 − 32t)j ' √ r(t) = v(t) dt = 240 3 ti + (240t − 16t2 )j + b 0 = r(0) = b. √ √ (a) The shell’s trajectory is given by r(t) = 240 3 ti + (240t − 16t2 )j or x = 240 3 t, y = 240t − 16t2 . (b) Solving dy/dt = 240 − 32t = 0, we see that y is maximum when t = 15/2. The maximum altitude is y(15/2) = 900 ft. (c) Solving y(t) = 240t − 16t2 = 16t(15 − t) = 0, we see that the shell is at ground level when √ t = 0 and t = 15. The range of the shell is x(15) = 3600 3 ≈ 6235 ft. (d) From (c), impact is when t = 15. The speed at impact is + √ ∥v(15)∥ = |240 3 i + (240 − 32 · 15)j| = 2402 · 3 + (−240)2 = 480 ft/s. √ 12. Initially we are given s0 = 1600j and v0 = (480 cos 30◦ )i + (480 sin 30◦ )j = 240 3 i + 240j. Using a(t) = −32j we find ' v(t) = a(t) dt = −32tj + c √ 240 3 i + 240j = v(0) = c √ √ v(t) = −32tj + 240 3 i + 240j = 240 3 i + (240 − 32t)j ' √ r(t) = v(t) dt = 240 3 ti + (240t − 16t2 )j + b (a) (b) 1600j = r(0) = b. √ √ The shell’s trajectory is given by r(t) = 240 3 ti + (240t − 16t2 + 1600)j or x = 240 3 t, y = 240t − 16t2 + 1600. Solving dy/dt = 240 − 32t = 0, we see that y is maximum when t = 15/2. altitude is y(15/2) = 2500 ft. The maximum Solving y(t) = −16t2 + 240t + 1600 = −16(t − 20)(t + 5) = 0, we see that the shell hits the ground √ when t = 20. The range of the shell is x(20) = 4800 3 ≈ 8314 ft. (d) From (c), impact is when t = 20. The speed at impact is + √ √ |v(20)∥ = |240 3 i + (240 − 32 · 20)j| = 2402 · 3 + (−400)2 = 160 13 ≈ 577 ft/s. (c) 13. We are given s0 = 81j and v0 = 4i. Using a(t) = −32j, we have ' v(t) = a(t) dt = −32tj + c 4i = v(0) = c r(t) = ' v(t) = 4i − 32tj v(t) dt = 4ti − 16t2 j + b 81j = r(0) = b r(t) = 4ti + (81 − 16t2 )j. 445 9.2 Motion on a Curve Solving y(t) = 81 − 16t2 = 0, we see that the car hits the water when t = 9/4. Then + √ ∥v(9/4)∥ = |4i − 32(9/4)j| = 42 + 722 = 20 13 ≈ 72.11 ft/s. 14. Let θ be the angle of elevation. Then v(0) = 98 cos θi + 98 sin θj. Using a(t) = −9.8j, we have ' v(t) = a(t) dt = −9.8tj + c 98 cos θi + 98 sin θj = v(0) = c v(t) = 98 cos θi + (98 sin θ − 9.8t)j r(t) = 98t cos θi + (98t sin θ − 4.9t2 )j + b. Since r(0) = 0, b = 0 and r(t) = 98t cos θi + (98t sin θ − 4.9t2 )j. Setting y(t) = 98t sin θ − 4.9t2 = t(98 sin θ − 4.9t) = 0, we see that the projectile hits the ground when t = 20 sin θ. Thus, using x(t) = 98t cos θ, 490 = x(t) = 98(20 sin θ) cos θ or sin 2θ = 0.5. Then 2θ = 30◦ or 150◦ . The angles of elevation are 15◦ and 75◦ . √ √ s 2 s 2 15. Let s be the initial speed. Then v(0) = s cos 45◦ i + s sin 45◦ j = i+ j. Using a(t) = −32j, we have 2 2 ' v(t) = a(t) dt = −32tj + c √ √ s 2 s 2 i+ j = v(0) = c 2 2 . / √ √ s 2 s 2 v(t) = i+ − 32t j 2 2 / . √ √ s 2 s 2 2 r(t) = ti + t − 16t j + b. 2 2 Since r(0) = 0, b = 0 and / . √ √ s 2 s 2 2 r(t) = ti + t − 16t j. 2 2 √ √ Setting y(t) = s 2 t/2 − 16t2 = t(s 2/2 − 16t) = 0 we see that the ball hits the √ Thus, using x(t) = s 2 t/2 and the fact that 100 yd = 300 ft, 300 = x(t) = √ s = 9600 ≈ 97.98 ft/s. √ ground when t = 2 s/32. √ s 2 √ s2 ( 2 s/32) = and 2 32 16. Let s be the initial speed and θ the initial angle. Then v(0) = s cos θi + s sin θj. Using a(t) = −32j, we have ' v(t) = a(t) dt = −32tj + c s cos θi + s sin θj = v(0) = c v(t) = s cos θi + (s sin θ − 32t)j r(t) = st cos θi + (st sin θ − 16t2 )j + b. Since r(0) = 0, b = 0 and r(t) = st cos θi + (st sin θ − 16t2 )j. Setting y(t) = st sin θ − 16t2 = t(s sin θ − 16t) = 0, we see that the ball hits the ground when t = (s sin θ)/16. Using x(t) = st cos θi, we see that the range of the ball is x ) s sin θ 16 * = s2 sin θ cos θ s2 sin 2θ = . 16 32 446 9.2 For θ = 30◦ , the range is s2 sin 60◦ /32 = √ Motion on a Curve 3 s2 /64 and for θ = 60◦ the range is s2 sin 120◦ /32 = √ 3 s2 /64. In general, when the angle is 90◦ − θ the range is [s2 sin 2(90◦ − θ)]/32 = s2 [sin(180◦ − 2θ)]/32 = s2 (sin 2θ)/32. Thus, for angles θ and 90◦ − θ, the range is the same. 17. Let the initial speed of the projectile be s and let the target be at (x0 , y0 ). Then vp (0) = s cos θi + s sin θj and vt (0) = 0. Using a(t) = −32j, we have ' vp (t) = a(t) dt = −32tj + c s cos θi + s sin θj = vp (0) = c vp (t) = s cos θi + (s sin θ − 32t)j rp (t) = st cos θi + (st sin θ − 16t2 )j + b. Since rp (0) = 0, b = 0 and rp (t) = st cos θi+(st sin θ−16t2 )j. Also, vt (t) = −32tj+c and since vt (0) = 0, c = 0 and vt (t) = −32tj. Then rt (t) = −16t2 j + b. Since rt (0) = x0 i + y0 j, b = x0 i + y0 j and rt (t) = x0 i + (y0 − 16t2 )j. Now, the horizontal component of rp (t) will be x0 when t = x0 /s cos θ at which time the vertical component of rp (t) will be (sx0 /s cos θ) sin θ − 16(x0 /s cos θ)2 = x0 tan θ − 16(x0 /s cos θ)2 = y0 − 16(x0 /s cos θ)2 . Thus, rp (x0 /s cos θ) = rt (x0 /s cos θ) and the projectile will strike the target as it falls. 18. The initial angle is θ = 0, the initial height is 1024 ft, and the initial speed is s = 180(5280)/3600 = 264 ft/s. Then x(t) = 264t and y(t) = −16t2 +1024. Solving y(t) = 0 we see that the pack hits the ground at t = 8 seconds The horizontal distance travelled is x(8) = 2112 feet. From the figure in the text, tan α = 1024/2112 = 16/33 and α ≈ 0.45 radian or 25.87◦ . 19. r′ (t) = v(t) = −r0 ω sin ωti + r0 ω cos ωtj; v = ∥v(t)∥ = , r02 ω 2 sin2 ωt + r02 ω 2 cos2 ωt = r0 ω ω = v/r0 ; a(t) = r′′ (t) = −r0 ω 2 cos ωti − r0 ω 2 sin ωtj , a = ∥a(t)∥ = r02 ω 4 cos2 ωt + r02 ω 4 sin2 ωt = r0 ω 2 = r0 (v/r0 )2 = v 2 /r0 . + √ 20. (a) v(t) = −b sin ti + b cos tj + ck; ∥v(t)∥ = b2 sin2 t + b2 cos2 t + c2 = b2 + c2 ' t+ ' t + ds + 2 = b + c2 ∥v(u)∥ du (b) s = b2 + c2 du = t b2 + c2 ; dt 0 0 + d2 s b2 cos2 t + b2 sin2 t = |b|. Thus, d2 s/dt2 ̸= ∥a(t)∥. = 0; a(t) = −b cos ti − b sin tj; ∥a(t)∥ = (c) dt2 21. By Problem 19, a = v 2 /r0 = 15302 /(4000 · 5280) ≈ 0.1108. We are given mg = 192, so m = 192/32 and we = 192 − (192/32)(0.1108) ≈ 191.33 lb. 22. By Problem 19, the centripetal acceleration is v 2 /r0 . Then the horizontal force is mv 2 /r0 . The vertical force is 32m. The resultant force is U = (mv 2 /r0 )i + 32mj. From the figure, we see that tan φ = (mv 2 /r0 )/32m = v 2 /32r0 . Using r0 = 60 and v = 44 we obtain tan φ = 442 /32(60) ≈ 1.0083 and φ ≈ 45.24◦ . 23. Solving x(t) = (v0 cos θ)t for t and substituting into y(t) = − 12 gt2 + (v0 sin θ)t + s0 we obtain ) *2 1 x x g y=− g + (v0 sin θ) + s0 = − 2 x2 + (tan θ)x + s0 , 2 v0 cos θ v0 cos θ 2v0 cos2 θ 447 9.2 Motion on a Curve which is the equation of a parabola. 24. Since the projectile is launched from ground level, s0 = 0. To find the maximum height we maximize y(t) = − 12 gt2 + (v0 sin θ)t. Solving y ′ (t) = −gt + v0 sin θ = 0, we see that t = (v0 /g) sin θ is a critical point. Since y ′′ (t) = −g < 0, ) * v0 sin θ 1 v 2 sin2 θ v0 sin θ v 2 sin2 θ H=y + v0 sin θ =− g 0 2 = 0 g 2 g g 2g is the maximum height. To find the range we solve y(t) = − 21 gt2 + (v0 sin θ)t = t(v0 sin θ − 12 gt) = 0. The positive solution of this equation is t = (2v0 sin θ)/g. The range is thus x(t) = (v0 cos θ) 2v0 sin θ v 2 sin 2θ = 0 . g g 25. Letting r(t) = x(t)i + y(t)j + z(t)k, the equation dr/dt = v is equivalent to dx/dt = 6t2 x, dy/dt = −4ty 2 , dz/dt = 2t(z + 1). Separating variables and integrating, we obtain dx/x = 6t2 dt, dy/y 2 = −4t dt, dz/(z + 1) = 2t dt, and ln x = 2t3 + c1 , −1/y = −2t2 + c2 , ln(z + 1) = t2 + c3 . Thus, 3 r(t) = k1 e2t i + 2 1 j + (k3 et − 1)k. 2t2 + k2 26. We require the fact that dr/dt = v. Then dL d dp dr = (r × p) = r × + × p = τ + v × p = τ + v × mv = τ + m(v × v) = τ + 0 = τ . dt dt dt dt 27. (a) Since F is directed along r we have F = cr for some constant c. Then τ = r × F = r × (cr) = c(r × r) = 0. (b) If τ = 0 then dL/dt = 0 and L is constant. 28. (a) Using Problem 27, F = −k(M m/r2 )u = ma. Then a = d2 r/dt = −k(M/r2 )u. (b) Using u = r/r we have * ! " kM kM 1 M r × r = r × −k 2 u = − 2 r × ( r) = − 3 (r × r) = 0. r r r r ′′ ) (c) From Theorem 9.4 (iv) we have d dv dr (r × v) = r × + × v = r × r′′ + v × v = 0 + 0 = 0. dt dt dt (d) Since r = ru we have c = r × v = ru × ru′ = r2 (u × u′ ). (e) Since u = (1/r)r is a unit vector, u · u = 1 and d du du du d (u · u) = u · + · u = 2u · = (1) = 0. dt dt dt dt dt Thus, u · u′ = 0. (f ) d dc dv kM kM (v × c) = v × + × c = v × 0 + a × c = − 2 u × c = − 2 u × [r2 (u × u′ )] dt dt dt r r ′ ′ = −kM [u × (u × u )] = −kM = −kM [(u · u )u − (u · u)u′ ] by (10) of 7.4 = −kM [0 − u′ ] = kM u′ = kM du dt 448 9.3 Curvature and Components of Acceleration (g) Since r · (v × c) = (r × v) · c = c · c = c2 by Problem 61 in 7.4 where c = ∥c∥ and (kM u + d) · r = (kM u + d) · ru = kM ru · u + rd · u = kM r + rd cos θ we have c2 = kM r + rd cos θ or r = where d = ∥d∥ c2 /kM c2 = . kM + d cos θ 1 + (d/kM ) cos θ (h) First note that c > 0 (otherwise there is no orbit) and d > 0 (since the orbit is not a circle). We recognize the equation in (g) to be that of a conic section with eccentricity e = d/kM . Since the orbit of the planet is closed it must be an ellipse. (i) At perihelion c = ∥c∥ = ∥r × v∥ = r0 v0 sin(π/r) = r0 v0 . Since r is minimum at this point, we want the denominator in the equation r0 = [c2 /kM ]/[1 + (d/kM ) cos θ] to be maximum. This occurs when θ = 0. In this case r2 v 2 /kM r0 = 0 0 and d = r0 v02 − kM. 1 + d/kM EXERCISES 9.3 Curvature and Components of Acceleration 1. r′ (t) = −t sin ti + t cos tj + 2tk; |r′ (t)| = cos t 2 sin t T(t) = − √ i + √ j + √ k 5 5 5 + t2 sin2 t + t2 cos2 t + 4t2 = 2. r′ (t) = et (− sin t + cos t)i + et (cos t + sin t)j + √ 2 et k, √ 5 t; √ |r′ (t)| = [e2t (sin2 t − 2 sin t cos t + cos2 t) + e2t (cos2 t + 2 sin t cos t + sin2 t) + 2e2t ]1/2 = 4e2t = 2et ; √ 1 2 1 k T(t) = (− sin t + cos t)i + (cos t + sin t)j + 2 2 2 + √ 3. We assume a > 0. r′ (t) = −a sin ti + a cos tj + ck; |r′ (t)| = a2 sin2 t + a2 cos2 t + c2 = a2 + c2 ; a cos t a sin t a cos t c dT a sin t T(t) = − √ = −√ i+ √ j+ √ k; i− √ j, 2 2 2 2 2 2 2 2 dt a +c a +c a +c a +c a2 + c2 ( 0 2 ( ( dT ( a cos2 t a2 sin2 t a ( ( + 2 =√ ; N = − cos ti − sin tj; ( dt ( = a2 + c2 a + c2 a2 + c2 ( ( i j k ( ( ( ( ( ( a cos t c a sin t ( = √c sin t i − √c cos t j + √ a √ √ B = T × N = (( − √ 2 k; ( 2 2 2 2 2 a + c a + c a + c a2 + c2 a2 + c2 a2 + c2 ( ( ( ( − cos t − sin t 0 √ |dT/dt| a/ a2 + c2 a √ κ= = 2 = |r′ (t)| a + c2 a2 + c2 449 9.3 Curvature and Components of Acceleration 4. r′ (t) = i + tj + t2 k, r′ (1) = i + j + k; |r′ (t)| = √ 1 + t2 + t4 , |r′ (1)| = √ 3; 1 T(t) = (1 + t2 + t4 )−1/2 (i + tj + t2 k), T(1) = √ (i + j + k); 3 dT t 1 = − (1 + t2 + t4 )−3/2 (2t + 4t3 )i + [(1 + t2 + t4 )−1/2 − (1 + t2 + t4 )−3/2 (2t + 4t3 )]j dt 2 2 t2 + [2t(1 + t2 + t4 )−1/2 − (1 + t2 + t4 )−3/2 (2t + 4t3 )]k; 2 √ ( ( ( ( 1 1 1 d d 1 1 2 T(1) = − √ i + √ k, (( T(1)(( = + = √ ; N(1) = − √ (i − k); dt dt 3 3 3 3 3 2 ( ( ( √ √ √ i j k (( ( ( ( ( ′ (d √ √ √ ( 2/ 3 2 1 ( ( ( B(1) = ( 1/ 3 1/ 3 1/ 3 ( = √ (i − 2j + k); κ = ( T(1)( /|r (1)| = √ = ( √ √ ( dt 3 6 3 ( −1/ 2 0 1/ 2 ( √ 5. From Example 2 in the text, a normal to the osculating plane is B(π/4) = √126 (3i−3j+2 2 k). The point on the √ √ √ √ √ curve when t = π/4 is ( 2 , 2 , 3π/4). An equation of the plane is 3(x − 2 ) − 3(y − 2 ) + 2 2(z − 3π/4) = 0, √ √ √ √ 3x − 3y + 2 2 z = 3 2 π/2, or 3 2 x − 3 2 y + 4z = 3π. 6. From Problem 4, a normal to the osculating plane is B(1) = √16 (i − 2j + k). The point on the curve when t = 1 is (1, 1/2, 1/3). An equation of the plane is (x − 1) − 2(y − 1/2) + (z − 1/3) = 0 or x − 2y + z = 1/3. √ 7. v(t) = j + 2tk, |v(t)| = 1 + 4t2 ; a(t) = 2k; v · a = 4t, v × a = 2i, |v × a| = 2; 4t 2 aT = √ , aN = √ 1 + 4t2 1 + 4t2 8. v(t) = −3 sin ti + 2 cos tj + k, + + + + |v(t)| = 9 sin2 t + 4 cos2 t + 1 = 5 sin2 t + 4 sin2 t + 4 cos2 t + 1 = 5 sin2 t + 1 ; a(t) = −3 cos ti − 2 sin tj; v · a = 9 sin t cos t − 4 sin t cos t = 5 sin t cos t, + + + v × a = 2 sin ti − 3 cos tj + 6k, |v × a| = 4 sin2 t + 9 cos2 t + 36 = 5 cos2 t + 8 ; 0 √ 5 sin t cos t cos2 t + 8 aT = + , aN = sin2 t + 1 sin2 t + 1 √ 9. v(t) = 2ti + 2tj + 4tk, |v(t)| = 2 6 t, t > 0; a(t) = 2i + 2j + 4k; v · a = 24t, v × a = 0; √ 24t aT = √ = 2 6 , aN = 0, t > 0 2 6t √ 10. v(t) = 2ti − 3t2 j + 4t3 k, |v(t)| = t 4 + 9t2 + 16t4 , t > 0; a(t) = 2i − 6tj + 12t2 k; √ v · a = 4t + 18t3 + 48t5 ; v × a = −12t4 i − 16t3 j − 6t2 k, |v × a| = 2t2 36t4 + 64t2 + 9 ; √ 4 + 18t2 + 48t4 2t 36t4 + 64t2 + 9 aT = √ , aN = √ ,t>0 4 + 9t2 + 16t4 4 + 9t2 + 16t4 √ 11. v(t) = 2i + 2tj, |v(t)| = 2 1 + t2 ; a(t) = 2j; v · a = 4t; v × a = 4k, |v × a| = 4; 2 2t , aN = √ aT = √ 1 + t2 1 + t2 √ 1 1 + t2 t 2t 1 − t2 12. v(t) = i + j, |v(t)| = ; a(t) = − i+ j; 2 2 2 2 2 1+t 1+t 1+t (1 + t ) (1 + t2 )2 v·a=− t − t3 t 1 1 2t + =− ; v×a= k, |v × a| = ; (1 + t2 )3 (1 + t2 )3 (1 + t2 )2 (1 + t2 )2 (1 + t2 )2 aT = − √ 1/(1 + t2 )2 t 1 t/(1 + t2 )2 √ , a = =− = N 2 3/2 2 2 2 2 (1 + t ) (1 + t2 )3/2 1 + t /(1 + t ) 1 + t /(1 + t ) 450 9.3 Curvature and Components of Acceleration 13. v(t) = −5 sin ti + 5 cos tj, |v(t)| = 5; a(t) = −5 cos ti − 5 sin tj; v · a = 0, v × a = 25k, |v × a| = 25; aT = 0, aN = 5 + 14. v(t) = sinh ti + cosh tj, |v(t)| = sinh2 t + cosh2 t ; a(t) = cosh ti + sinh tj v · a = 2 sinh t cosh t; 1 2 sinh t cosh t , aN = + v × a = (sinh2 t − cosh2 t)k = −k, |v × a| = 1; aT = + 2 2 2 sinh t + cosh t sinh t + cosh2 t √ 15. v(t) = −e−t (i + j + k), |v(t)| = 3 e−t ; a(t) = e−t (i + j + k); v · a = −3e−2t ; v × a = 0, |v × a| = 0; √ −t aT = − 3 e , aN = 0 √ 16. v(t) = i + 2j + 4k, |v(t)| = 21 ; a(t) = 0; v · a = 0, v × a = 0, |v × a| = 0; aT = 0, aN = 0 + 17. v(t) = −a sin ti + b cos tj + ck, |v(t)| = a2 sin2 t + b2 cos2 t + c2 ; a(t) = −a cos ti − b sin tj; + v × a = bc sin ti − ac cos tj + abk, |v × a| = b2 c2 sin2 t + a2 c2 cos2 t + a2 b2 + b2 c2 sin2 t + a2 c2 cos2 t + a2 b2 |v × a| ; = κ= |v|3 (a2 sin2 t + b2 cos2 t + c2 )3/2 + 18. (a) v(t) = −a sin ti + b cos tj, |v(t)| = a2 sin2 t + b2 cos2 t ; a(t) = −a cos ti − b sin tj; ab v × a = abk; |v × a| = ab; κ = 2 2 (a sin t + b2 cos2 t)3/2 (b) When a = b, |v(t)| = a, |v × a| = a2 , and κ = a2 /a3 = 1/a. 19. The equation of a line is r(t) = b + tc, where b and c are constant vectors. v(t) = c, |v(t)| = |c|; a(t) = 0; v × a = 0, |v × a| = 0; κ = |v × a|/|v|3 = 0 20. v(t) = a(1 − cos t)i + a sin tj; v(π) = 2ai, |v(π)| = 2a; a(t) = a sin ti + a cos tj, a(π) = −aj; ( ( ( i j k (( ( |v × a| 2a2 1 ( ( |v × a| = ( 2a 0 0 ( = −2a2 k; |v × a| = 2a2 ; κ = = = 3 3 ( ( |v| 8a 4a ( 0 −a 0 ( + 21. v(t) = f ′ (t)i + g ′ (t)j, |v(t)| = [f ′ (t)]2 + [g ′ (t)]2 ; a(t) = f ′′ (t)i + g ′′ (t)j; v × a = [f ′ (t)g ′′ (t) − g ′ (t)f ′′ (t)]k, |v × a| = |f ′ (t)g ′′ (t) − g ′ (t)f ′′ (t)|; κ= |v × a| |f ′ (t)g ′′ (t) − g ′ (t)f ′′ (t)| = 3 |v| ([f ′ (t)]2 + [g ′ (t)]2 )3/2 22. For y = F (x), r(x) = xi + F (x)j. We identify f (x) = x and g(x) = F (x) in Problem 21. Then f ′ (x) = 1, f ′′ (x) = 0, g ′ (x) = F ′ (x), g ′′ (x) = F ′′ (x), and κ = |F ′′ (x)|/(1 + [F ′ (x)]2 )3/2 . 23. F (x) = x2 , F (0) = 0, F (1) = 1; F ′ (x) = 2x, F ′ (0) = 0, F ′ (1) = 2; F ′′ (x) = 2, F ′′ (0) = 2, F ′′ (1) = 2; 2 2 1 2 κ(0) = = 2; ρ(0) = ; κ(1) = = √ ≈ 0.18; 2 3/2 2 3/2 2 (1 + 0 ) (1 + 2 ) 5 5 √ √ 5 5 ρ(1) = ≈ 5.59; Since 2 > 2/5 5 , the curve is “sharper” at (0, 0). 2 24. F (x) = x3 , F (−1) = −1, F (1/2) = 1/8; F ′ (x) = 3x2 , F ′ (−1) = 3, F ′ (1/2) = 3/4; F ′′ (x) = 6x, F ′′ (−1) = −6, F ′′ (1/2) = 3; κ(−1) = 3 6 | − 6| = √ = √ ≈ 0.19; (1 + 32 )3/2 10 10 5 10 √ 3 5 10 1 3 192 1 125 = ρ(−1) = ≈ 5.27; κ( ) = = ≈ 1.54; ρ( ) = ≈ 0.65 2 3/2 3 2 125/64 125 2 192 [1 + (3/4) ] Since 1.54 > 0.19, the curve is “sharper” at (1/2, 1/8). 25. At a point of inflection (x0 , F (x0 )), if F ′′ (x0 ) exists then F ′′ (x0 ) = 0. Thus, assuming that limx→x0 F ′′ (x) exists, F ′′ (x) and hence κ is near 0 for x near x0 . 451 9.3 Curvature and Components of Acceleration 26. We use the fact that T · N = 0 and T · T = N · N = 1. Then |a(t)|2 = a · a = (aN N + aT T) · (aN N + aT T) = a2N N · N + 2aN aT N · T + a2T T · T = a2N + a2T . EXERCISES 9.4 Partial Derivatives 1. y = − 12 x + C 2. x = y 2 − c 3. x2 − y 2 = 1 + c2 4. 4x2 + 9y 2 = 36 − c2 , −6 ≤ c ≤ 6 5. y = x2 + ln c, c > 0 6. y = x + tan c, −π/x < c < π/2 7. x2 /9 + z 2 /4 = c; elliptical cylinder 8. x2 + y 2 + z 2 = c; sphere 9. x2 + 3y 2 + 6z 2 = c; ellipsoid 10. 4y − 2z + 1 = c; plane 11. 452 9.4 Partial Derivatives 12. Setting x = −4, y = 2, and z = −3 in x2 /16 + y 2 /4 + z 2 /9 = c we obtain c = 3. The equation of the surface is √ x2 /16 + y 2 /4 + z 2 /9 = 3. Setting y = z = 0 we find the x-intercepts are ±4 3 . Similarly, the y-intercepts are √ √ ±2 3 and the z-intercepts are ±3 3 . 13. zx = 2x − y 2 ; zy = −2xy + 20y 4 14. zx = −3x2 + 12xy 3 ; zy = 18x2 y 2 + 10y 15. zx = 20x3 y 3 − 2xy 6 + 30x4 ; zy = 15x4 y 2 − 6x2 y 5 − 4 16. zx = 3x2 y 2 sec2 (x3 y 2 ); zy = 2x3 y sec2 (x3 y 2 ) √ 24y x 2 ; z = − 17. zx = √ y (3y 2 + 1)2 x (3y 2 + 1) 18. zx = 12x2 − 10x + 8; zy = 0 19. zx = −(x3 − y 2 )−2 (3x2 ) = −3x2 (x3 − y 2 )−2 ; zy = −(x3 − y 2 )−2 (−2y) = 2y(x3 − y 2 )−2 20. zx = 6(−x4 + 7y 2 + 3y)5 (−4x3 ) = −24x3 (−x4 + 7y 2 + 3y)5 ; zy = 6(−x4 + 7y 2 + 3y)5 (14y + 3) 21. zx = 2(cos 5x)(− sin 5x)(5) = −10 sin 5x cos 5x; zy = 2(sin 5y)(cos 5y)(5) = 10 sin 5y cos 5y 2 22. zx = (2x tan−1 y 2 )ex 3 tan−1 y 2 ; zy = 3 2x2 y x2 tan−1 y2 e 1 + y4 3 3 23. fx = x(3x2 yex y + ex y = (3x3 y + 1)ex y ; fy = x4 ex y ) *) * *) ) * 1 θ θ θ θ θ θ θ 24. fθ = φ2 cos = φ cos ; fφ = φ2 cos − 2 + 2φ sin = −θ cos + 2φ sin φ φ φ φ φ φ φ φ 25. fx = 7y (x + 2y)(−1) − (3x − y)(2) −7x (x + 2y)3 − (3x − y) = ; fy = = (x + 2y)2 (x + 2y)2 (x + 2y)2 (x + 2y)2 26. fx = −3x2 y − y 3 (x2 − y 2 )2 y − xy[2(x2 − y 2 )2x] = ; 2 2 4 (x − y ) (x2 − y 2 )3 fy = (x2 − y 2 )2 x − xy[2(x2 − y 2 )(−2y)] 3xy 2 + x3 = (x2 − y 2 )4 (x2 − y 2 )3 15v 2 8u ; g = v 4u2 + 5v 3 4u2 + 5v 3 √ √ 1 1 s r 28. hr = √ + 2 ; hs = − 2 − √ r s 2s r 2r s ) * $y $ y % y2 % √ √ 1 y/z y 29. wx = √ ; wy = 2 x − y e + 1 ey/z ; wz = −yey/z − 2 = 2 ey/z − ey/z = 2 x − z z z z x ) * xy 1 + (ln xz)y = y + y ln xz; wy = x ln xz; wz = 30. wx = xy x z 27. gu = 31. Fu = 2uw2 − v 3 − vwt2 sin(ut2 ); Fv = −3uv 2 + w cos(ut2 ); Fx = 4(2x2 t)3 (4xt) = 16xt(2x2 t)3 = 128x7 t4 ; Ft = −2uvwt sin(ut2 ) + 64x8 t3 32. Gp = r4 s5 (p2 q 3 )r 4 5 Gq = r4 s5 (p2 q 3 )r 4 5 Gs = (p2 q 3 )r 33. 4 5 s s −1 s −1 (2pq 3 ) = 2pq 3 r4 s5 (p2 q 3 )r 4 5 s −1 (3p2 q 2 ) = 3p2 q 2 r4 s5 (p2 q 3 )r 4 5 ; s −1 ; Gr = (p2 q 3 )r 4 5 s (4r3 s5 ) ln(p2 q 3 ); (5r4 s4 ) ln(p2 q 3 ) ∂z ∂2z (x2 + y 2 )2 − 2x(2x) 2y 2 − 2x2 ∂z 2x 2y , = = ; , = 2 = 2 2 2 2 2 2 2 2 2 ∂x x +y ∂x (x + y ) (x + y ) ∂y x + y2 ∂2z (x2 + y 2 )2 − 2y(2y) 2x2 − 2y 2 ∂2z ∂2z 2y 2 − 2x2 + 2x2 − 2y 2 = = 2 ; + 2 = =0 2 2 2 2 2 2 2 ∂y (x + y ) (x + y ) ∂x ∂y (x2 + y 2 )2 453 9.4 34. Partial Derivatives 2 2 2 2 ∂z = ex −y (−2y sin 2xy) + 2xex −y cos 2xy ∂x 2 2 ∂2z = ex −y (−4y 2 cos 2xy − 8xy sin 2xy + 4x2 cos 2xy + 2 cos 2xy) 2 ∂x 2 2 2 2 ∂z = ex −y (−2x sin 2xy) − 2yex −y cos 2xy ∂y 2 2 ∂2z = ex −y (−4x2 cos 2xy + 8xy sin 2xy + 4y 2 cos 2xy − 2 cos 2xy) ∂y 2 Adding the second partial derivatives gives ∂2z ∂2z + = [−4(y 2 + x2 ) cos 2xy + 4(x2 + y 2 ) cos 2xy] = 0. ∂x2 ∂y 2 35. 36. 37. ∂u ∂2u ∂u ∂2u = cos at cos x, = − cos at sin x; = −a sin at sin x, = −a2 cos at sin x; ∂x ∂x2 ∂t ∂t2 ∂2u ∂2u a2 2 = a2 (− cos at sin x) = 2 ∂x ∂t ∂u ∂2u = − sin(x + at) + cos(x − at), = − cos(x + at) − sin(x − at); ∂x ∂x2 ∂2u ∂u = −a2 cos(x + at) − a2 sin(x − at); = −a sin(x + at) − a cos(x − at), ∂t ∂t2 ∂2u ∂2u a2 2 = −a2 cos(x + at) − a2 sin(x − at) = 2 ∂x ∂t 2 ∂C 2x ∂2C 4x2 −1/2 −x2 /kt 2 −1/2 −x2 /kt = − t−1/2 e−x /kt , = t e − t e ; 2 2 2 ∂x kt ∂x k t kt 2 ∂C x2 t−3/2 −x2 /kt k ∂ 2 C x2 −1/2 −x2 /kt t−1/2 −x2 /kt ∂C ; = t e − = = t−1/2 2 e−x /kt − e e ∂t kt 2 4 ∂x2 kt2 2t ∂t 38. (a) Pv = −k(T /V 2 ) (b) P V = kt, P VT = k, VT = k/P (c) P V = kT , V = kTp , Tp = V /k 2 2 2 2 2 2 39. zx = v 2 euv (3x2 ) + 2uveuv (1) = 3x2 v 2 euv + 2uveuv ; zy = v 2 euv (0) + 2uveuv (−2y) = −4yuveuv 40. zx = (2u cos 4v)(2xy 3 ) − (4u2 sin 4v)(3x2 ) = 4xy 3 u cos 4v − 12x2 u2 sin 4v zy = (2u cos 4v)(3x2 y 2 ) − (4v 2 sin 4v)(3y 2 ) = 6x2 y 2 u cos 4v − 12y 2 u2 sin 4v 41. zu = 4(4u3 ) − 10y[2(2u − v)(2)] = 16u3 − 40(2u − v)y zv = 4(−24v 2 ) − 10y[2(2u − v)(−1)] = −96v 2 + 20(2u − v)y ) * ) 2* 2y −2x 2xv 2 1 v 2y 42. zu = + + 2 − 2 = 2 2 2 (x + y) v (x + y) u v(x + y) u (x + y)2 * ) $ u% −2x 4xv 2v 2yu 2y − 2 + =− 2 − zv = 2 (x + y) v (x + y)2 u v (x + y)2 u(x + y)2 3 2 3 (u + v 2 )1/2 (2u)(−e−t sin θ) + (u2 + v 2 )1/2 (2v)(−e−t cos θ) 2 2 2 2 1/2 −t 2 = −3u(u + v ) e sin θ − 3v(u + v 2 )1/2 e−t cos θ 3 3 wθ = (u2 + v 2 )1/2 (2u)e−t cos θ + (u2 + v 2 )1/2 (2v)(−e−t sin θ) 2 2 2 2 1/2 −t = 3u(u + v ) e cos θ − 3v(u2 + v 2 )1/2 e−t sin θ 43. wt = 454 2 9.4 Partial Derivatives √ √ rs2 u v/2 uv u/2 uv rv +√ (2r) + (2rs2 ) = √ 1 + uv 1 + uv uv (1 + uv) uv (1 + uv) √ √ v/2 uv u/2 uv −sv r2 su ws = +√ (−2s) + (2r2 s) = √ 1 + uv 1 + uv uv (1 + uv) uv (1 + uv) 44. wr = 2 2 45. Ru = s2 t4 (ev ) + 2rst4 (−2uve−u ) + 4rs2 t3 (2uv 2 eu 2 2 Rv = s2 t4 (2uvev ) + 2rst4 (e−u ) + 4rs2 t3 (2u2 veu t2 1 − x2 ) * 1 1 + + q t2 * ) 1 2x 1 −1 Qt = (2t sin x) + − 3 + p q t 1 46. Qx = P ) √ * 2 2 2 2 v v 2 2 ) = s2 t4 ev − 4uvrst4 e−u + 8uv 2 rs2 t3 eu 2 2 ) = 2s2 t4 uvev + 2rst4 e−u + 8rs2 t3 u2 veu 1 r ) 1/t 1 + (x/t)2 * 1 r ) −x/t2 1 + (x/t)2 t t2 1 = √ + 2+ 2 + x2 ) 2 qt r(t p 1−x * = 2 2 v 2 2 v 2x x 2t sin−1 x − 3− p qt r(t2 + x2 ) 2x u cosh rs y cosh rs 2y xu 47. wt = + + + + + =+ u 2 x2 + y 2 rs + tu 2 x2 + y 2 x2 + y 2 (rs + tu) u x2 + y 2 yst sinh rs s st sinh rs 2x 2y xs + + wr = + + + =+ u 2 x2 + y 2 rs + tu 2 x2 + y 2 x2 + y 2 (rs + tu) u x2 + y 2 2y xt 2x yt cosh rs t −t cosh rs + + =+ wu = + − + 2 u 2 x2 + y 2 rs + tu 2 x2 + y 2 x2 + y 2 (rs + tu) u2 x2 + y 2 48. sφ = 2pe3θ + 2q[− sin(φ + θ)] − 2rθ2 + 4(2) = 2pe3θ − 2q sin(φ + θ) − 2rθ2 + 8 sθ = 2p(3φe3θ ) + 2q[− sin(φ + θ)] − 2r(2φθ) + 4(8) = 6pφe3θ − 2q sin(φ + θ) − 4rφθ + 32 49. 2u 2v 4ut − 4vt−3 dz = 2 (2t) + 2 (−2t−3 ) = 2 2 dt u +v u +v u2 + v 2 50. dz = (3u2 v − v 4 )(−5e−5t ) + (u3 − 4uv 3 )(5 sec 5t tan 5t) = −5(3u2 v − v 4 )e−5t + 5(u3 − 4uv 3 ) sec 5t tan 5t dt 51. dw dt dw dt = −3 sin(3u + 4v)(2) − 4 sin(3u + 4v)(−1); u(π) = 5π/2, v(π) = −5π/4 ( * ) * ) ( ( = −6 sin 15π − 5π + 4 sin 15π − 5π = −2 sin 5π = −2 ( 2 2 2 π ( ! " dw dw (( −8 xy 52. = 5e20 (−8) + 4e20 (3) = −28e20 + xe (3); x(0) = 4, y(0) = 5; = yexy dt (2t + 1)2 dt (0 53. With x = r cos θ and y = r sin θ ∂u ∂u ∂x ∂u ∂y ∂u ∂u = + = cos θ + sin θ ∂r ∂x ∂r ∂y ∂r ∂x ∂y ∂2u ∂ 2 u ∂x ∂ 2 u ∂y ∂2u ∂2u 2 = cos θ + sin θ = cos θ + sin2 θ ∂r2 ∂x2 ∂r ∂y 2 ∂r ∂x2 ∂y 2 ∂u ∂x ∂u ∂y ∂u ∂u ∂u = + = (−r sin θ) + (r cos θ) ∂θ ∂x ∂θ ∂y ∂θ ∂x ∂y ∂2u ∂u ∂ 2 u ∂x ∂u ∂ 2 u ∂y = (−r cos θ) + (−r sin θ) + (−r sin θ) + 2 (r cos θ) 2 2 ∂θ ∂x ∂x ∂θ ∂y ∂y ∂θ = −r ∂2u ∂u ∂2u ∂u cos θ + r2 2 sin2 θ − r sin θ + r2 2 cos2 θ. ∂x ∂x ∂y ∂y 455 9.4 Partial Derivatives Using ∂2u ∂2u + 2 = 0, we have ∂x2 ∂y 1 ∂2u ∂2u ∂2u ∂ 2 u 1 ∂u + = cos2 θ + 2 sin2 θ + + 2 2 2 2 ∂r r ∂r r ∂θ ∂x ∂y ) ∂u 1 cos θ + r2 + 2 −r r ∂x 1 r ) ∂u ∂u cos θ + sin θ ∂x ∂y * * 2 ∂2u ∂u 2 2 ∂ u 2 sin θ − r sin θ + r cos θ ∂x2 ∂y ∂y 2 * ) ∂2u ∂2u ∂u 1 1 2 2 2 2 = (cos θ + sin θ) + 2 (sin θ + cos θ) + cos θ − cos θ ∂x2 ∂y ∂x r r ) * ∂u 1 1 sin θ − sin θ + ∂y r r = 54. ∂2u ∂2u + 2 = 0. ∂x2 ∂y dP (V − 0.0427)(0.08)dT /dt 0.08T (dV /dt) 3.6 dV − + 3 = dt (V − 0.0427)2 (V − 0.0427)2 V dt * ) 3.6 0.08T dV dT 0.08 = − + V − 0.0427 dt V3 (V − 0.0427)2 dt 55. Since dT /dT = 1 and ∂P/∂T = 0, 0 = FT = ∂F ∂P ∂F ∂V ∂F dT ∂V ∂F/∂T 1 + + =⇒ =− =− . ∂P ∂T ∂V ∂T ∂T dT ∂T ∂F/∂V ∂T /∂V 56. We are given dE/dt = 2 and dR/dt = −1. Then when E = 60 and R = 50, dI ∂I dE ∂I dR 1 E = + = (2) − 2 (−1), dt ∂E dt ∂R dt R R dI 1 2 60 3/5 8 = + = + = amp/min. dt 50 502 25 25 125 57. Since the height of the triangle is x sin θ, the area is given by A = 12 xy sin θ. Then 1 dx 1 dy 1 dθ dA ∂A dx ∂A dy ∂A dθ + + = y sin θ + x sin θ + xy cos θ . dt ∂x dt ∂y dt ∂θ dt 2 dt 2 dt 2 dt When x = 10, y = 8, θ = π/6, dx/dt = 0.3, dy/dt = 0.5, and dθ/dt = 0.1, .√ / ) * ) * 1 1 1 1 1 3 dA = (8) (0.3) + (10) (0.5) + (10)(8) (0.1) dt 2 2 2 2 2 2 √ √ = 0.6 + 1.25 + 2 3 = 1.85 + 2 3 ≈ 5.31 cm2 /s. 58. −4x sin t + 4y cos t + 5z ∂w dx ∂w dy ∂w dz x dx/dt + y dy/dt + z dz/dt dw + =+ = + + = dt ∂x dt ∂y dt ∂z dt x2 + y 2 + z 2 16 cos2 t + 16 sin2 t + 25t2 = −16 sin t cos t + 16 sin t cos t + 25t 25t √ =√ 2 16 + 25t 16 + 25t2 ( 125π/2 dw (( 125π =+ ≈ 4.9743 =√ 2 dt (t=5π/2 64 + 625π 2 16 + 625π /4 456 and 9.5 Directional Derivative EXERCISES 9.5 Directional Derivative 1. ∇f = (2x − 3x2 y 2 )i + (4y 3 − 2x3 y)j 2 2 2. ∇f = 4xye−2x y i + (1 + 2x2 e−2x y )j 2xy 3xy 2 y2 i + j − k z3 z3 z4 4. ∇F = y cos yzi + (x cos yz − xyz sin yz)j − xy 2 sin yzk 3. ∇F = 5. ∇f = 2xi − 8yj; ∇f (2, 4) = 4i − 32j 3x2 x3 − 4y 3 27 5 6. ∇f = + i+ + j; ∇f (3, 2) = √ i − √ j 3 4 3 4 38 2 38 2 x y−y 2 x y−y 7. ∇F = 2xz 2 sin 4yi + 4x2 z 2 cos 4yj + 2x2 z sin 4yk √ √ 4π 4π 4π ∇F (−2, π/3, 1) = −4 sin i + 16 cos j + 8 sin k = 2 3 i − 8j − 4 3 k 3 3 3 2y 2z 4 3 1 2x 8. ∇F = 2 i+ 2 j+ 2 k; ∇F (−4, 3, 5) = − i + j+ k x + y2 + z2 x + y2 + z2 x + y2 + z2 25 25 5 √ √ (x + h 3/2)2 + (y + h/2)2 − x2 − y 2 f (x + h 3/2, y + h/2) − f (x, y) 9. Du f (x, y) = lim = lim h→0 h→0 h h √ 2 2 √ √ h 3 x + 3h /4 + hy + h /4 = lim = lim ( 3 x + 3h/4 + y + h/4) = 3 x + y h→0 h→0 h √ √ √ √ f (x + h 2/2, y + h 2/2) − f (x, y) 3x + 3h 2/2 − (y + h 2/2)2 − 3x + y 2 10. Du f (x, y) = lim = lim h→0 h→0 h h √ √ 2 √ √ √ √ 3h 2/2 − h 2 y − h /2 = lim = lim (3 2/2 − 2 y − h/2) = 3 2/2 − 2 y h→0 h→0 h √ 1 3 11. u = i + j; ∇f = 15x2 y 6 i + 30x3 y 5 j; ∇f (−1, 1) = 15i − 30j; 2 2 √ 15 3 15 √ Du f (−1, 1) = − 15 = ( 3 − 2) 2 2 √ √ 2 2 12. u = i+ j; ∇f = (4 + y 2 )i + (2xy − 5)j; ∇f (3, −1) = 5i − 11j; 2 2 √ √ √ 5 2 11 2 Du f (3, −1) = − = −3 2 2 2 √ √ 3 10 −y x 1 1 10 i− j; ∇f = 2 i+ 2 j; ∇f (2, −2) = i + j 13. u = 10 10 x + y2 x + y2 4 4 √ √ √ 10 3 10 10 Du f (2, −2) = − =− 40 40 20 8 3 4 y2 x2 6 i+ j = i + j; ∇f = i + j; ∇f (2, −1) = i + 4j 10 10 5 5 (x + y)2 (x + y)2 3 16 19 Du f (2, −1) = + = 5 5 5 14. u = 457 9.5 Directional Derivative √ 15. u = (2i + j)/ 5 ; ∇f = 2y(xy + 1)i + 2x(xy + 1)j; ∇f (3, 2) = 28i + 42j 42 98 2(28) Du f (3, 2) = √ + √ = √ 5 5 5 16. u = −i; ∇f = 2x tan yi + x2 sec2 yj; ∇f (1/2, π/3) = √ √ 3 i + j; Du f (1/2, π/3) = − 3 1 1 17. u = √ j + √ k; ∇F = 2xy 2 (2z + 1)2 i + 2x2 y(2z + 1)2 j + 4x2 y 2 (2z + 1)k 2 2 √ 18 12 6 ∇F (1, −1, 1) = 18i − 18j + 12k; Du F (1, −1, 1) = − √ + √ = − √ = −3 2 2 2 2 1 2y 2y 2 − 2x2 2 1 2x 18. u = √ i − √ j + √ k; ∇F = 2 i − 2 j + k; ∇F (2, 4, −1) = 4i − 8j − 24k z z z3 6 6 6 √ 16 24 4 Du F (2, 4, −1) = √ − √ − √ = −6 6 6 6 6 19. u = −k; ∇F = + x2 + 4z y2 i+ + j+ + k x2 y + 2y 2 z 2 x2 y + 2y 2 z x2 y + 2y 2 z xy ∇F (−2, 2, 1) = −i + j + k; Du F (−2, 2, 1) = −1 √ 2 2 1 20. u = −(4i − 4j + 2k)/ 36 = − i + j − k; ∇F = 2i − 2yj + 2zk; ∇F (4, −4, 2) = 2i + 8j + 4k 3 3 3 8 4 16 4 − = Du F (4, −4, 2) = − + 3 3 3 3 √ 12 16 4 21. u = (−4i − j)/ 17 ; ∇f = 2(x − y)i − 2(x − y)j; ∇f (4, 2) = 4i − 4j; Du F (4, 2) = − √ √ = − √ 17 17 17 √ 22. u = (−2i + 5j)/ 29 ; ∇f = (3x2 − 5y)i − (5x − 2y)j; ∇f (1, 1) = −2i − 3j; 15 11 4 Du f (1, 1) = √ − √ = − √ 29 29 29 √ √ 2 2x 2x 23. ∇f = 2e sin yi + e cos yj; ∇f (0, π/4) = 2 i + j 2 + √ √ √ √ 2 The maximum Du is [( 2 ) + ( 2/2)2 ]1/2 = 5/2 in the direction 2 i + ( 2/2)j. 24. ∇f = (xyex−y + yex−y )i + (−xyex−y + xex−y )j; ∇f (5, 5) = 30i − 20j √ The maximum Du is [302 + (−20)2 ]1/2 = 10 13 in the direction 30i − 20j. 25. ∇F = (2x + 4z)i + 2z 2 j + (4x + 4yz)k; ∇F (1, 2, −1) = −2i + 2j − 4k √ The maximum Du is [(−2)2 + 22 + (−4)2 ]1/2 = 2 6 in the direction −2i + 2j − 4k. 26. ∇F = yzi + xzj + xyk; ∇F (3, 1, −5) = −5i − 15j + 3k √ The maximum Du is [(−5)2 + (−15)2 + 32 ]1/2 = 259 in the direction −5i − 15j + 3k. 27. ∇f = 2x sec2 (x2 + y 2 )i + 2y sec2 (x2 + y 2 )j; + + + + ∇f ( π/6 , π/6 ) = 2 π/6 sec2 (π/3)(i + j) = 8 π/6 (i + j) + + The minimum Du is −8 π/6 (12 + 12 )1/2 = −8 π/3 in the direction −(i + j). 28. ∇f = 3x2 i − 3y 2 j; ∇f (2, −2) = 12i − 12j = 12(i − j) √ The minimum Du is −12[12 + (−1)2 ]1/2 = −12 2 in the direction −(i − j) = −i + j. √ √ y √ ze x 2 3 y 29. ∇F = √ i + xz e j + √ k; ∇F (16, 0, 9) = i + 12j + k. The minimum Du is 8 3 2 x 2 z √ 3 2 −[(3/8)2 + 122 + (2/3)2 ]1/2 = − 83,281/24 in the direction − i − 12j − k. 8 3 458 9.5 Directional Derivative 1 1 1 i + j − k; ∇F (1/2, 1/6, 1/3) = 2i + 6j − 3k x y z The minimum Du is −[22 + 62 + (−3)2 ]1/2 = −7 in the direction −2i − 6j + 3k. 30. ∇F = 31. Using implicit differentiation on 2x2 + y 2 = 9 we find y ′ = −2x/y. At (2, 1) the slope of the tangent line is √ −2(2)/1 = −4. Thus, u = ±(i − 4j)/ 17 . Now, ∇f = i + 2yj and ∇f (3, 4) = i + 8j. Thus, √ √ √ Du = ±(1/ 17 − 32 17 ) = ±31/ 17 . 3x + 3y − 1 2x + y − 1 x + 2y √ √ + √ = 32. ∇f = (2x + y − 1)i + (x + 2y)j; Du f (x, y) = 2 2 2 √ Solving (3x + 3y − 1)/ 2 = 0 we see that Du is 0 for all points on the line 3x + 3y = 1. * ) 3 4 i− j . 33. (a) Vectors perpendicular to 4i + 3j are ±(3i − 4j). Take u = ± 5 5 √ 4 3 (b) u = (4i + 3j)/ 16 + 9 = i + j 5 5 3 4 (c) u = − i − j 5 5 34. D−u f (a, b) = ∇f (a, b) · (−u) = −∇f (a, b) · u = −Du f (a, b) = −6 35. (a) ∇f = (3x2 − 6xy 2 )i + (−6x2 y + 3y 2 )j Du f (x, y) = 9x2 − 18xy 2 − 6x2 y + 3y 2 3(3x2 − 6xy 2 ) −6x2 y + 3y 2 √ √ √ + = 10 10 10 3 3 (b) F (x, y) = √ (3x2 − 6xy 2 − 2x2 y + y 2 ); ∇F = √ [(6x − 6y 2 − 4xy)i + (−12xy − 2x2 + 2y)j] 10 10 ) ) *) * *) * 3 3 1 3 2 √ √ √ √ Du F (x, y) = (6x − 6y − 4xy) + (−12xy − 2x2 + 2y) 10 10 10 10 9 3 1 2 2 = (3x − 3y − 2xy) + (−6xy − x + y) = (27x − 27y 2 − 36xy − 3x2 + 3y) 5 5 5 Gmy Gm Gmx i+ 2 j= 2 (xi + yj) 36. ∇U = 2 (x + y 2 )3/2 (x + y 2 )3/2 (x + y 2 )3/2 The maximum and minimum values of Du U (x, y) are obtained when u is in the directions ∇U and −∇U , respectively. Thus, at a point (x, y), not (0, 0), the directions of maximum and minimum increase in U are xi + yj and −xi − yj, respectively. A vector at (x, y) in the direction ±(xi + yj) lies on a line through the origin. 37. ∇f = (3x2 − 12)i + (2y − 10)j. Setting |∇f | = [(3x2 − 12)2 + (2y − 10)2 ]1/2 = 0, we obtain 3x2 − 12 = 0 and 2y − 10 = 0. The points where |∇f | = 0 are (2, 5) and (−2, 5). 38. Let ∇f (a, b) = αi + βj. Then Du f (a, b) = ∇f (a, b) · u = 5 12 5 12 α − β = 7 and Dv f (a, b) = ∇f (a, b) · v = α − β = 3. 13 13 13 13 Solving for α and β, we obtain α = 13 and β = −13/6. Thus, ∇f (a, b) = 13i − (13/6)j. 39. ∇T = 4xi + 2yj; ∇T (4, 2) = 16i + 4j. The minimum change in temperature (that is, the maximum decrease in temperature) is in the direction −∇T (4, 3) = −16i − 4j. 40. Let x(t)i + y(t)j be the vector equation of the path. At (x, y) on this curve, the direction of a tangent vector is x′ (t)i + y ′ (t)j. Since we want the direction of motion to be −∇T (x, y), we have x′ (t)i + y ′ (t)j = −∇T (x, y) = 4xi + 2yj. Separating variables in dx/dt = 4x, we obtain dx/x = 4 dt, ln x = 4t + c1 , and x = C1 e4t . Separating variables in dy/dt = 2y, we obtain dy/y = 2 dt, ln y = 2t + c2 , and y = C2 e2t . Since x(0) = 4 and y(0) = 2, we 459 9.5 Directional Derivative have x = 4e4t and y = 2e2t . The equation of the path is 4e4t i + 2e2t j for t ≥ 0, or eliminating the parameter, x = y 2 , y ≥ 0. 41. Let x(t)i + y(t)j be the vector equation of the path. At (x, y) on this curve, the direction of a tangent vector is x′ (t)i + y ′ (t)j. Since we want the direction of motion to be ∇T (x, y), we have x′ (t)i + y ′ (t)j = ∇T (x, y) = −4xi − 2yj. Separating variables in dx/dt = −4x we obtain dx/x = −4 dt, ln x = −4t + c1 and x = C1 e−4t . Separating variables in dy/dt = −2y we obtain dy/y = −2 dt, ln y = −2t + c2 and y = C2 e−2t . Since x(0) = 3 and y(0) = 4, we have x = 3e−4t and y = 4e−2t . The equation of the path is 3e−4t i + 4e−2t j, or eliminating the parameter, 16x = 3y 2 , y ≥ 0. 42. Substituting x = 0, y = 0, z = 1, and T = 500 into T = k/(x2 + y 2 + z 2 ) we see that k = 500 and T (x, y, z) = 500/(x2 + y 2 + z 2 ) . To find the rate of change of T at ⟨2, 3, 3⟩ in the direction of ⟨3, 1, 1⟩ we first compute ⟨3, 1, 1⟩ − ⟨2, 3, 3⟩ = ⟨1, −2, −2⟩. Then u = 13 ⟨1, −2, −2⟩ = 13 i − 32 j − 23 k. Now ∇T = − so (x2 1000y 1000z 500 750 750 1000x i− 2 j− 2 k and ∇T (2, 3, 3) = − i− j− k, 2 2 2 2 2 2 2 2 2 +y +z ) (x + y + z ) (x + y + z ) 121 121 121 1 Du T (2, 3, 3) = 3 ) 500 − 121 * 2 − 3 ) 750 − 121 * 2 − 3 ) 750 − 121 750 750 The direction of maximum increase is ∇T (2, 3, 3) = − 500 121 i− 121 j− 121 k = √ √ 250 rate of change of T is |∇T (2, 3, 3)| = 250 22. 121 4 + 9 + 9 = 121 * = 2500 . 363 250 121 (−2i−3j−3k), and the maximum 43. Since ∇f = fx (x, y)i + fy (x, y)j, we have ∂f /∂x = 3x2 + y 3 + yexy . Integrating, we obtain f (x, y) = x3 + xy 3 + exy + g(y). Then fy = 3xy 2 + xexy + g ′ (y) = −2y 2 + 3xy 2 + xexy . Thus, g ′ (y) = −2y 2 , g(y) = − 23 y 3 + c, and f (x, y) = x3 + xy 3 + exy − 32 y 3 + C. 44. Let u = u1 i + u2 j and v = v1 i + v2 j. Dv f = (fx i + fy j) · v = v1 fx + v2 fy ! " ∂ ∂ Du Dv f = (v1 fx + v2 fy )i + (v1 fx + v2 fy )j · u = [(v1 fxx + v2 fyx )i + (v1 fxy + v2 fyy )j] · u ∂x ∂y = u1 v1 fxx + u1 v2 fyx + u2 v1 fxy + u2 v2 fyy Du f = (fx i + fy j) · u = u1 fx + u2 fy " ! ∂ ∂ (u1 fx + u2 fy )i + (u1 fx + u2 fy )j · v = [(u1 fxx + u2 fyx )i + (u1 fxy + u2 fyy )j] · v Dv Du f = ∂x ∂y = u1 v1 fxx + u2 v1 fyx + u1 v2 fxy + u2 v2 fyy Since the second partial derivatives are continuous, fxy = fyx and Du Dv f = Dv Du f . [Note that this result is a generalization of fxy = fyx since Di Dj f = fyx and Dj Di f = fxy .] ∂ ∂ (cf )i + (cf )j = cfx i + cfy j = c(fx i + fy j) = c∇f ∂x ∂y 46. ∇(f + g) = (fx + gx )i + (fy + gy )j = (fx i + fy j) + (gx i + gy j) = ∇f + ∇g 45. ∇(cf ) = 47. ∇(f g) = (f gx + fx g)i + (f gy + fy g)j = f (gx i + gy j) + g(fx i + fy j) = f ∇g + g∇f 48. ∇(f /g) = [(gfx − f gx )/g 2 ]i + [(gfy − f gy )/g 2 ]j = g(fx i + fy j)/g 2 − f (gx i + gy j)/g 2 = g∇f /g 2 − f ∇g/g 2 = (g∇f − f ∇g)/g 2 ( ( ( i j k (( ) * ) * ) * ( ∂f1 ∂f2 ∂f2 ∂f3 ∂f1 ∂f3 ( ( − i+ − j+ − k 49. ∇ × F = ( ∂/∂x ∂/∂y ∂/∂z ( = ( ( ∂y ∂z ∂z ∂x ∂x ∂y ( f1 ( f2 f3 460 9.6 Tangent Planes and Normal Lines EXERCISES 9.6 Tangent Planes and Normal Lines 1. Since f (6, 1) = 4, the level curve is x − 2y = 4. ∇f = i − 2j; ∇f (6, 1) = i − 2j 2. Since f (1, 3) = 5, the level curve is y + 2x = 5x or y = 3x, x ̸= 0. y 1 ∇f = − 2 i + j; ∇f (1, 3) = −3i + j x x 3. Since f (2, 5) = 1, the level curve is y = x2 + 1. ∇f = −2xi + j; ∇f (2, 5) = −10i + j 4. Since f (−1, 3) = 10, the level curve is x2 + y 2 = 10. ∇f = 2xi + 2yj; ∇f (−1, 3) = −2i + 6j 5. Since f (−2, −3) = 2, the level curve is x2 /4 + y 2 /9 = 2 x 2y 2 or x2 /8 + y 2 /18 = 1. ∇f = i + j; ∇f (−2, −3) = −i − j 2 9 3 6. Since f (2, 2) = 2, the level curve is y 2 = 2x, x ̸= 0. ∇f = − ∇f (2, 2) = −i + 2j y2 2y i + j; x2 x 7. Since f (1, 1) = −1, the level curve is (x − 1)2 − y 2 = −1 or y 2 − (x − 1)2 = 1. ∇f = 2(x − 1)i − 2yj; ∇f (1, 1) = −2j 8. Since f (π/6, 3/2) = 1, the level curve is y − 1 = sin x or y = 1 + sin x, sin x ̸= 0. √ −(y − 1) cos x 1 ∇f = i+ j; ∇f (π/6, 3/2) = − 3 i + 2j 2 sin x sin x 9. Since F (3, 1, 1) = 2, the level surface is y + z = 2. ∇F = j + k; ∇F (3, 1, 1) = j + k 461 9.6 Tangent Planes and Normal Lines 10. Since F (1, 1, 3) = −1, the level surface is x2 + y 2 − z = −1 or z = 1 + x2 + y 2 . ∇F = 2xi + 2yj − k; ∇F (1, 1, 3) = 2i + 2j − k 11. Since F (3, 4, 0) = 5, the level surface is x2 + y 2 + z 2 = 25. y z x ∇F = + i+ + j+ + k; 2 2 2 2 2 2 2 x +y +z x +y +z x + y2 + z2 3 4 ∇F (3, 4, 0) = i + j 5 5 12. Since F (0, −1, 1) = 0, the level surface is x2 − y 2 + z = 0 or z = y 2 − x2 . ∇F = 2xi − 2yj + k; ∇F (0, −1, 1) = 2j + k 13. F (x, y, z) = x2 + y 2 − z; ∇F = 2xi + 2yj − k. We want ∇F = c(4i + j + 21 k) or 2x = 4c, 2y = c, −1 = c/2. From the third equation c = −2. Thus, x = −4 and y = −1. Since z = x2 + y 2 = 16 + 1 = 17, the point on the surface is (−4, −1, −17). 14. F (x, y, z) = x3 + y 2 + z; ∇F = 3x2 i + 2yj + k. We want ∇F = c(27i + 8j + k) or 3x2 = 27c, 2y = 8c, 1 = c. From c = 1 we obtain x = ±3 and y = 4. Since z = 15 − x3 − y 2 = 15 − (±3)3 − 16 = −1 ∓ 27, the points on the surface are (3, 4, −28) and (−3, 4, 26). 15. F (x, y, z) = x2 + y 2 + z 2 ; ∇F = 2xi + 2yj + 2zk. ∇F (−2, 2, 1) = −4i + 4j + 2k. The equation of the tangent plane is −4(x + 2) + 4(y − 2) + 2(z − 1) = 0 or −2x + 2y + z = 9. 16. F (x, y, z) = 5x2 − y 2 + 4z 2 ; ∇F = 10xi − 2yj + 8zk; ∇F (2, 4, 1) = 20i − 8j + 8k. The equation of the tangent plane is 20(x − 2) − 8(y − 4) + 8(z − 1) = 0 or 5x − 2y + 2z = 4. 17. F (x, y, z) = x2 − y 2 − 3z 2 ; ∇F = 2xi − 2yj − 6zk; ∇F (6, 2, 3) = 12i − 4j − 18k. The equation of the tangent plane is 12(x − 6) − 4(y − 2) − 18(z − 3) = 0 or 6x − 2y − 9z = 5. 18. F (x, y, z) = xy + yz + zx; ∇F = (y + z)i + (x + z)j + (y + x)k; ∇F (1, −3, −5) = −8i − 4j − 2k. The equation of the tangent plane is −8(x − 1) − 4(y + 3) − 2(z + 5) = 0 or 4x + 2y + z = −7. 19. F (x, y, z) = x2 + y 2 + z; ∇F = 2xi + 2yj + k; ∇F (3, −4, 0) = 6i − 8j + k. The equation of the tangent plane is 6(x − 3) − 8(y + 4) + z = 0 or 6x − 8y + z = 50. 20. F (x, y, z) = xz; ∇F = zi+xk; ∇F (2, 0, 3) = 3i+2k. The equation of the tangent plane is 3(x−2)+2(z −3) = 0 or 3x + 2z = 12. √ √ √ 2 21. F (x, y, z) = cos(2x + y) − z; ∇F = −2 sin(2x + y)i − sin(2x + y)j − k; ∇F (π/2, π/4, −1/ 2 ) = 2 i + j − k. 2 √ $ * ) $ % % √ π 2 π 1 = 0, The equation of the tangent plane is 2 x − + y− − z+√ 2 2 4 2 * ) $ √ π% $ π% √ 1 5π 2 x− + y− − 2 z+√ = 0, or 2x + y − 2 z = + 1. 2 4 4 2 22. F (x, y, z) = x2 y 3 + 6z; ∇F = 2xy 3 i + 3x2 y 2 j + 6k; ∇F (2, 1, 1) = 4i + 12j + 6k. The equation of the tangent plane is 4(x − 2) + 12(y − 1) + 6(z − 1) = 0 or 2x + 6y + 3z = 13. 462 9.6 Tangent Planes and Normal Lines √ √ √ √ 2y 2x i+ 2 j − k; ∇F (1/ 2 , 1/ 2 , 0) = 2 i + 2 j − k. 2 2 +y x +y * * ) ) √ √ 1 1 + 2 y− √ − (z − 0) = 0, The equation of the tangent plane is 2 x − √ 2 2 ) * ) * √ √ √ 1 1 2 x− √ +2 y− √ − 2 z = 0, or 2x + 2y − 2 z = 2 2 . 2 2 √ 24. F (x, y, z) = 8e−2y sin 4x − z; ∇F = 32e−2y cos 4xi − 16e−2y sin 4xj − k; ∇F (π/24, 0, 4) = 16 3 i − 8j − k. The equation of the tangent plane is √ √ √ 2 3π 16 3(x − π/24) − 8(y − 0) − (z − 4) = 0 or 16 3 x − 8y − z = − 4. 3 23. F (x, y, z) = ln(x2 + y 2 ) − z; ∇F = x2 25. The gradient of F (x, y, z) = x2 + y 2 + z 2 is ∇F = 2xi + 2yj + 2zk, so the normal vector to the surface at (x0 , y0 , z0 ) is 2x0 i + 2y0 j + 2z0 k. A normal vector to the plane 2x + 4y + 6z = 1 is 2i + 4j + 6k. Since we want the tangent plane to be parallel to the given plane, we find c so that 2x0 = 2c, 2y0 = 4c, 2z0 = 6c or x0 = c, √ y0 = 2c, z0 = 3c. Now, (x0 , y0 , z0 ) is on the surface, so c2 + (2c)2 + (3c)2 = 14c2 = 7 and c = ±1/ 2 . Thus, √ √ √ √ √ √ the points on the surface are ( 2/2, 2 , 3 2/2) and − 2/2, − 2 , −3 2/2). 26. The gradient of F (x, y, z) = x2 − 2y 2 − 3z 2 is ∇F (x, y, z) = 2xi − 4yj − 6zk, so a normal vector to the surface at (x0 , y0 , z0 ) is ∇F (x0 , y0 , z0 ) = 2x0 i−4y0 j−6z0 k. A normal vector to the plane 8x+4y+6z = 5 is 8i+4j+6k. Since we want the tangent plane to be parallel to the given plane, we find c so that 2x0 = 8c, −4y0 = 4c, −6z0 = 6c or x0 = 4c, y0 = −c, z0 = −c. Now, (x0 , y0 , z0 ) is on the surface, so (4c)2 − 2(−c)2 − 3(−c)2 = 11c2 = 33 and √ √ √ √ √ √ √ c = ± 3 . Thus, the points on the surface are (4 3 , − 3 , − 3) and (−4 3 , 3 , 3 ). 27. The gradient of F (x, y, z) = x2 + 4x + y 2 + z 2 − 2z is ∇F = (2x + 4)i + 2yj + (2z − 2)k, so a normal to the surface at (x0 , y0 , z0 ) is (2x0 + 4)i + 2y0 j + (2z0 − 2)k. A horizontal plane has normal ck for c ̸= 0. Thus, we want 2x0 + 4 = 0, 2y0 = 0, 2z0 − 2 = c or x0 = −2, y0 = 0, z0 = c + 1. since (x0 , y0 , z0 ) is on the surface, (−2)2 + 4(−2) + (c + 1)2 − 2(c + 1) = c2 − 5 = 11 and c = ±4. The points on the surface are (−2, 0, 5) and (−2, 0, −3). 28. The gradient of F (x, y, z) = x2 + 3y 2 + 4z 2 − 2xy is ∇F = (2x − 2y)i + (6y − 2x)j + 8zk, so a normal to the surface at (x0 , y0 , z0 ) is 2(x0 − y0 )i + 2(3y0 − x0 )j + 8z0 k. (a) A normal to the xz plane is cj for c ̸= 0. Thus, we want 2(x0 − y0 ) = 0, 2(3y0 − x0 ) = c, 8z0 = 0 or x0 = y0 , 3y0 − x0 = c/2, z0 = 0. Solving the first two equations, we obtain x0 = y0 = c/4. Since (x0 , y0 , z0 ) is on √ the surface, (c/4)2 + 3(c/4)2 + 4(0)2 − 2(c/4)(c/4) = 2c2 /16 = 16 and c = ±16/ 2 . Thus, the points on √ √ √ √ the surface are (4/ 2 , 4/ 2 , 0) and (−4/ 2 , −4/ 2 , 0). (b) A normal to the yz-plane is ci for c ̸= 0. Thus, we want 2(x0 − y0 ) = c, 2(3y0 − x0 ) = 0, 8z0 = 0 or x0 − y0 = c/2, x0 = 3y0 , z0 = 0. Solving the first two equations, we obtain x0 = 3c/4 and y0 = c/4. Since (x0 , y0 , z0 ) √ is on the surface, (3c/4)2 + 3(c/4)2 + 4(0)2 − 2(3c/4)(c/4) = 6c2 /16 = 16 and c = ±16 6 . Thus, the points √ √ √ √ √ √ on the surface are (12/ 6 , 4/ 6 , 0) on the surface are (12/ 6 , 4/ 6 , 0) and (−12/ 6 , −4/ 6 , 0). (c) A normal to the xy-plane is ck for c ̸= 0. Thus, we want 2(x0 − y0 ) = 0, 2(3y0 − x0 ) = 0, 8z0 = c or x0 = y0 , 3y0 − x0 = 0, z0 = c/8. Solving the first two equations, we obtain x0 = y0 = 0. Since (x0 , y0 , z0 ) is on the surface, 02 + 3(0)2 + 4(c/8)2 − 2(0)(0) = c2 /16 = 16 and c = ±16. Thus, the points on the surface are (0, 0, 2) and (0, 0, −2). 29. If (x0 , y0 , z0 ) is on x2 /a2 + y 2 /b2 + z 2 /c2 = 1, then x20 /a2 + y02 /b2 + z02 /c2 = 1 and (x0 , y0 , z0 ) is on the plane xx0 /a2 + yy0 /b2 + zz0 /c2 = 1. A normal to the surface at (x0 , y0 , z0 ) is ∇F (x0 , y0 , z0 ) = (2x0 /a2 )i + (2y0 /b2 )j + (2z0 /c2 )k. 463 9.6 Tangent Planes and Normal Lines A normal to the plane is (x0 /a2 )i + (y0 /b2 )j + (z0 /c2 )k. Since the normal to the surface is a multiple of the normal to the plane, the normal vectors are parallel and the plane is tangent to the surface. 30. If (x0 , y0 , z0 ) is on x2 /a2 − y 2 /b2 + z 2 /c2 = 1, then x20 /a2 − y02 /b2 + z02 /c2 = 1 and (x0 , y0 , z0 ) is on the plane xx0 /a2 − yy0 /b2 + zz0 /c2 = 1. A normal to the surface at (x0 , y0 , z0 ) is ∇F (x0 , y0 , z0 ) = (2x0 /a2 )i − (2y0 /b2 )j + (2z0 /c2 )k. A normal to the plane is (x0 /a2 )i − (y0 /b2 )j + (z0 /c2 )k. Since the normal to the surface is a multiple of the normal to the plane, the normal vectors are parallel, and the plane is tangent to the surface. 31. Let F (x, y, z) = x2 + y 2 − z 2 . Then ∇F = 2xi + 2yj − 2zk and a normal to the surface at (x0 , y0 , z0 ) is x0 i + y0 j − z0 k. An equation of the tangent plane at (x0 , y0 , z0 ) is x0 (x − x0 ) + y0 (y − y0 ) − z0 (z − z0 ) = 0 or x0 x + y0 y − z0 z = x20 + y02 − z02 . Since (x0 , y0 , z0 ) is on the surface, z02 = x20 + y02 and x20 + y02 − z02 = 0. Thus, the equation of the tangent plane is x0 x + y0 y − z0 z = 0, which passes through the origin. √ √ 1 1 1 √ 32. Let F (x, y, z) = x + y + z . Then ∇F = √ i + √ j + √ k and a normal to the surface at (x0 , y0 , z0 ) 2 y 2 x 2 z 1 1 1 is √ i + √ j + √ k. An equation of the tangent plane at (x0 , y0 , z0 ) is 2 x0 2 y0 2 z0 1 1 1 √ (x − x0 ) + √ (y − y0 ) + √ (z − z0 ) = 0 2 x0 2 y0 2 z0 or √ √ √ 1 1 1 √ √ x + √ y + √ z = x0 + y0 + z0 = a . x0 y0 z0 √ √ √ √ √ √ √ √ √ √ √ √ The sum of the intercepts is x0 a + y0 a + z0 a = ( x0 + y0 + z0 ) a = a · a = a. 33. F (x, y, z) = x2 + 2y 2 + z 2 ; ∇F = 2xi + 4yj + 2zk; ∇F (1, −1, 1) = 2i − 4j + 2k. Parametric equations of the line are x = 1 + 2t, y = −1 − 4t, z = 1 + 2t. 34. F (x, y, z) = 2x2 − 4y 2 − z; ∇F = 4xi − 8yj − k; ∇F (3, −2, 2) = 12i + 16j − k. Parametric equations of the line are x = 3 + 12t, y = −2 + 16t, z = 2 − t. 35. F (x, y, z) = 4x2 + 9y 2 − z; ∇F = 8xi + 18yj − k; ∇F (1/2, 1/3, 3) = 4i + 6j − k. Symmetric equations of the y − 1/3 z−3 x − 1/2 = = . line are 4 6 −1 36. F (x, y, z) = x2 + y 2 − z 2 ; ∇F = 2xi + 2yj − 2zk; ∇F (3, 4, 5) = 6i + 8j − 10k. Symmetric equations of the line y−4 z−5 x−3 = = . are 6 8 −10 37. A normal to the surface at (x0 , y0 , z0 ) is ∇F (x0 , y0 , z0 ) = 2x0 i+2y0 j+2z0 k. Parametric equations of the normal line are x = x0 + 2x0 t, y = y0 + 2y0 t, z = z0 + 2z0 t. Letting t = −1/2, we see that the normal line passes through the origin. 38. The normal lines to F (x, y, z) = 0 and G(x, y, z) = 0 are Fx i + Fy j + Fz k and Gx i + Gy j + Gz k, respectively. These vectors are orthogonal if and only if their dot product is 0. Thus, the surfaces are orthogonal at P if and only if Fx Gx + Fy Gy + Fz Gz = 0. 39. Let F (x, y, z) = x2 + y 2 + z 2 − 25 and G(x, y, z) = −x2 + y 2 + z 2 . Then Fx Gx + Fy Gy + Fz gz = (2x)(−2x) + (2y)(2y) + (2z)(2z) = 4(−x2 + y 2 + z 2 ). For (x, y, z) on both surfaces, F (x, y, z) = G(x, y, z) = 0. Thus, Fx Gx + Fy Gy + Fz Gz = 4(0) = 0 and the surfaces are orthogonal at points of intersection. 464 9.7 Divergence and Curl 40. Let F (x, y, z) = x2 − y 2 + z 2 − 4 and G(x, y, z) = 1/xy 2 − z. Then Fx Gx + Fy Gy + Fz Gz = (2x)(−1/x2 y 2 ) + (−2y)(−2/xy 3 ) + (2z)(−1) = −2/xy 2 + 4/xy 2 − 2z = 2(1/xy 2 − z). For (x, y, z) on both surfaces, F (x, y, z) = G(x, y, z) = 0. Thus, Fx Gx + Fy Gy + Fz Gz = 2(0) and the surfaces are orthogonal at points of intersection. EXERCISES 9.7 Divergence and Curl y 1. y 2. 3 3 x -3 x 3 -3 -3 -3 y 3. 3 y 4. 3 3 x -3 x 3 -3 -3 -3 y 5. 3 y 6. 3 3 x -3 x 3 -3 -3 3 -3 465 9.7 Divergence and Curl 7. curl F = (x − y)i + (x − y)j; div F = 2z 8. curl F = −2x2 i + (10y − 18x2 )j + (4xz − 10z)k; div F = 0 9. curl F = 0; div F = 4y + 8z 10. curl F = (xe2y + ye−yz + 2xye2y )i − ye2y j + 3(x − y)2 k; div F = 3(x − y)2 − ze−yz 11. curl F = (4y 3 − 6xz 2 )i + (2z 3 − 3x2 )k; div F = 6xy 2 1 12. curl F = −x3 zi + (3x2 yz − z)j + 23 x2 y 2 − y − 15y 2 k; div F = (x3 y − x) − (x3 y − x) = 0 13. curl F = (3e−z − 8yz)i − xe−z j; div F = e−z + 4z 2 − 3ye−z yz − 3z + 3xy 2 z 2 x 15. curl F = (xy 2 ey + 2xyey + x3 yez + x3 yzez )i − y 2 ey j + (−3x2 yzez − xex )k; div F = xyex + yex − x3 zez 14. curl F = (2xyz 3 + 3y)i + (y ln x − y 2 z 3 )j + (2 − z ln x)k; div F = 16. curl F = (5xye5xy + e5xy + 3xz 3 sin xz 3 − cos xz 3 )i + (x2 y cos yz − 5y 2 e5xy )j + (−z 4 sin xz 3 − x2 z cos yz)k; div F = 2x sin yz 17. div r = 1 + 1 + 1 = 3 ( ( ( i j k (( ( ( ( 18. curl r = ( ∂/∂x ∂/∂y ∂/∂z ( = 0i − 0j + 0k = 0 ( ( ( x y z ( ( ( ( i j k (( ) * ) * ) * ( ∂ ∂ ∂ ∂ ∂ ∂ ( ( − a3 i + a3 − a1 j + a1 − a2 k 19. a × ∇ = ( a1 a2 a3 ( = a2 ( ( ∂z ∂y ∂x ∂z ∂y ∂x ( ∂/∂x ∂/∂y ∂/∂z ( ( ( ( ( i j k ( ( ( ∂ ∂ ∂ ∂ ∂ (( (a × ∇) × r = (( a2 − a3 a3 − a1 a1 − a2 ∂x ∂z ∂y ∂x (( ( ∂z ( ( x y z 20. 21. 22. 23. = (−a1 − a1 )i − (a2 + a2 )j + (−a3 − a3 )k = −2a * ) ∂ ∂ ∂ + a2 + a3 r = 3a − (a1 i + a2 j + a3 k) = 2a ∇ × (a × r) = (∇ · r)a − (∇ · a)r = (1 + 1 + 1)a − a1 ∂x ∂y ∂z ( ( ( ∂/∂x ∂/∂y ∂/∂z ( ( ( ∂ ∂ ∂ ( ( a2 a3 ( = (a2 z − a3 y) − (a1 z − a3 x) + (a1 y − a2 x) = 0 ∇ · (a × r) = ( a1 ( ∂x ( ∂y ∂z ( ( x y z ( ( ( i j k (( ( ( ( ∇ × r = ( ∂/∂x ∂/∂y ∂/∂z ( = 0; a × (∇ × r) = a × 0 = 0 ( ( ( x y z ( ( ( ( i j k (( ( ( ( r × a = ( x y z ( = (a3 y − a2 z)i − (a3 x − a1 z)j + (a2 x − a1 y)k; r · r = x2 + y 2 + z 2 ( ( ( a1 a2 a3 ( ( ( ( ( i j k ( ( ( ( ∇ × [(r · r)a] = ( ∂/∂x ∂/∂y ∂/∂z ( ( ( ( (r · r)a1 (r · r)a2 (r · r)a3 ( = (2ya3 − 2za2 )i − (2xa3 − 2za1 )j + (2xa2 − 2ya1 )k = 2(r × a) 24. r · a = a1 x + a2 y + a3 z; r · r = x2 + y 2 + z 2 ; ∇ · [(r · r)a] = 2xa1 + 2ya2 + 2za3 = 2(r · a) 466 9.7 Divergence and Curl 25. Let F = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k and G = S(x, y, z)i + T (x, y, z)j + U (x, y, z)k. ∇ · (F + G) = ∇ · [(P + S)i + (Q + T )j + (R + U )k] = Px + Sx + Qy + Ty + Rz + Uz = (Px + Qy + Rz ) + (Sx + Ty + Uz ) = ∇ · F + ∇ · G 26. Let F = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k and G = S(x, y, z)i + T (x, y, z)j + U (x, y, z)k. ( ( ( i j k (( ( ( ( ∂/∂z ( ∇ × (F + G) = ( ∂/∂x ∂/∂y ( ( (P + S Q + T R + U ( = (Ry + Uy − Qz − Tz )i − (Rx + Ux − Pz − Sz )j + (Qx + Tx − Py − Sy )k = (Ry − Qz )i − (Rx − Pz )j + (Qx − Py )k + (Uy − Tz )i − (Ux − Sz )j + (Tx − Sy )k =∇×F+∇×G 27. ∇ · (f F) = ∇ · (f P i + f Qj + f Rk) = f Px + P fx + f Qy + Qfy + f Rz + Rfz = f (Px + Qy + Rz ) + (P fx + Qfy + Rfz ) = f (∇ · F) + F · (∇f ) ( ( ( i j k (( ( ( ( 28. ∇ × (f F) = ( ∂/∂x ∂/∂y ∂/∂z ( ( ( ( fP fQ fR ( = (f Ry + Rfy − f Qz − Qfz )i − (f Rx + Rfx − f Pz − P fz )j + (f Qx + Qfx − f Py − P fy )k = (f Ry − f Qz )i − (f Rx − f Pz )j + (f Qx − f Py )k + (Rfy − Qfz )i − (Rfx − P fz )j + (Qfx − P fy )k ( ( i ( ( = f [(Ry − Qz )i − (Rx − Pz )j + (Qx − Py )k + ( fx ( (P 29. Assuming continuous second partial derivatives, j fy Q ( k (( ( fz ( = f (∇ × F) + (∇f ) × F ( R( ( ( i ( ( curl (grad f ) = ∇ × (fx i + fy j + fz k) = ( ∂/∂x ( ( fx j ∂/∂y fy ( k (( ( ∂/∂z ( ( fz ( = (fzy − fyz )i − (fzx − fxz )j + (fyx − fxy )k = 0. 30. Assuming continuous second partial derivatives, div (curl F) = ∇ · [(Ry − Qz )i − (Rx − Pz )j + (Qx − Py )k] = (Ryx − Qzx − (Rxy − Pzy ) + (Qxz − Pyz ) = 0. 31. Let F = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k and G = S(x, y, z)i + T (x, y, z)j + U (x, y, z)k. ( ( ( i j k( ( ( ( ( F × G = ( P Q R ( = (QU − RT )i − (P U − RS)j + (P T − QS)k ( ( (S T U( div (F × G) = (QUx + Qx U − RTx − Rx T ) − (P Uy + Py U − RSy − Ry S) + (P Tz + Pz T − QSz − Qz S) = S(Ry − Qz ) + T (Pz − Rx ) + U (Qx − Py ) − P (Uy − Tz ) − Q(Sz − Ux ) − R(Tx − Sy ) = G · (curl F) − F · (curl G) 32. Using Problems 26 and 29, curl (curl F + grad f ) = ∇ × (curl F + grad f ) = ∇ × (curl F) + ∇ × (grad f ) = curl (curl F) + curl (grad f ) = curl (curl F) + 0 = curl (curl F). 467 9.7 Divergence and Curl 33. ∇ · ∇f = ∇ · (fx i + fy j + fz k) = fxx + fyy + fzz 34. Using Problem 27, ∇ · (f ∇f ) = f (∇ · ∇f ) + ∇f · ∇f = f (∇2 f ) + |∇f |2 . 35. curl F = −8yzi − 2zj − xk; curl (curl F) = 2i − (8y − 1)j + 8zk 36. (a) For F = P i + Qj + Rk, curl (curl F) = (Qxy − Pyy − Pzz + Rxz )i + (Ryz − Qzz − Qxx + Pyx )j + (Pzx − Rxx − Ryy + Qzy )k and −∇2 F + grad (div F) = −(Pxx + Pyy + Pzz )i − (Qxx + Qyy + Qzz )j − (Rxx + Ryy + Rzz )k + grad (Px + Qy + Rz ) = −Pxx i − Qyy j − Rzz k + (−Pyy − Pzz )i + (−Qxx − Qzz )j + (−Rxx − Ryy )k + (Pxx + Qyx + Rzx )i + (Pxy + Qyy + Rzy )j + (Pxz + Qyz + Rzz )k = (−Pyy − Pzz + Qyx + Rzx )i + (−Qxx − Qzz + Pxy + Rzy )j + (−Rxx − Ryy + Pxz + Qyz )k. Thus, curl (curl F) = −∇2 F + grad (div F). (b) For F = xyi + 4yz 2 j + 2xzk, ∇2 F = 0i + 8yj + 0k, div F = y + 4z 2 + 2x, and grad (div F) = 2i + j + 8zk. Then curl (curl F) = −8yj + 2i + j + 8zk = 2i + (1 − 8y)j + 8zk. 37. ∂f = −x(x2 + y 2 + z 2 )−3/2 ∂x ∂f = −y(x2 + y 2 + z 2 )−3/2 ∂y ∂f = −z(x2 + y 2 + z 2 )−3/2 ∂z ∂2f = 3x2 (x2 + y 2 + z 2 )−5/2 − (x2 + y 2 + z 2 )−3/2 ∂x2 ∂2f = 3y 2 (x2 + y 2 + z 2 )−5/2 − (x2 + y 2 + z 2 )−3/2 ∂y 2 ∂2f = 3z 2 (x2 + y 2 + z 2 )−5/2 − (x2 + y 2 + z 2 )−3/2 ∂z 2 Adding the second partial derivatives gives ∂2f ∂2f 3(x2 + y 2 + z 2 ) ∂2f − 3(x2 + y 2 + z 2 )−3/2 + 2 + 2 = 2 2 ∂x ∂y ∂z (x + y 2 + z 2 )5/2 = 3(x2 + y 2 + z 2 )−3/2 − 3(x2 + y 2 + z 2 )−3/2 = 0 except when x = y = z = 0. * ) 1 4xy 4xy 38. fx = =− 2 − (x2 + y 2 − 1)2 (x + y 2 − 1)2 + 4y 2 4y 2 1+ 2 (x + y 2 − 1)2 [(x2 + y 2 − 1)2 + 4y 2 ]4y − 4xy[4x(x2 + y 2 − 1)] 12x4 y − 4y 5 + 8x2 y 3 − 8x2 y − 8y 3 − 4y = [(x2 + y 2 − 1)2 + 4y 2 ]2 [(x2 + y 2 − 1)2 + 4y 2 ]2 ! " 2(x2 − y 2 − 1) 2(x2 + y 2 − 1) − 4y 2 1 = fy = (x2 + y 2 − 1)2 (x2 + y 2 − 1)2 + 4y 2 4y 2 1+ 2 2 2 (x + y − 1) fxx = − 468 9.7 fyy = = Divergence and Curl [(x2 + y 2 − 1)2 + 4y 2 ](−4y) − 2(x2 − y 2 − 1)[4y(x2 + y 2 − 1) + 8y] [(x2 + y 2 − 1)2 + 4y 2 ]2 −12x4 y + 4y 5 − 8x2 y 3 + 8x2 y + 8y 3 + 4y [(x2 + y 2 − 1)2 + 4y 2 ]2 ∇2 f = fxx + fyy = 0 ( ( i ( ( 39. curl F = −Gm1 m2 ( ∂/∂x ( ( x/|r|3 j ( ( ( ( ∂/∂z ( ( z/|r|3 ( k ∂/∂y y/|r|3 = −Gm1 m2 [(−3yz/|r|5 + 3yz/|r|5 )i − (−3xz/|r|5 + 3xz/|r|5 )j + (−3xy/|r|5 + 3xy/|r|5 )k] =0 " −2x2 + y 2 + z 2 x2 − 2y 2 + z 2 x2 + y 2 − 2z 2 + + =0 |r|5/2 |r|5/2 |r|5/2 ( ( ( ( i j k ( ( 1 1( 1 ( ω × r) = ( ∂/∂x 40. curl v = curl (ω ∂/∂y ∂/∂z ( ( 2 2 2( ( ω2 z − ω3 y ω 3 x − ω1 z ω1 y − ω2 x ( 1 = [(ω1 + ω1 )i − (−ω2 − ω2 )j + (ω3 + ω3 )k] = ω1 i + ω2 j + ω3 k = ω 2 div F = −Gm1 m2 ! 41. Using Problems 31 and 29, ∇ · F = div (∇f × ∇g) = ∇g · (curl ∇f ) − ∇f · (curl ∇g) = ∇g · 0 − ∇f · 0 = 0. 42. Recall that a · (a × b) = 0. Then, using Problems 31, 29, and 28, ∇ · F = div (∇f × f ∇g) = f ∇g · (curl ∇f ) − ∇f · (curl f ∇g) = f ∇g · 0 − ∇f · (∇ × f ∇g) = −∇f · [f (∇ × ∇g) + (∇f × ∇g)] = −∇f · [f curl ∇g + (∇f × ∇g] = −∇f · [f 0 + (∇f × ∇g)] = −∇f · (∇f × ∇g) = 0. 43. (a) Expressing the vertical component of V in polar coordinates, we have 2xy 2r2 sin θ cos θ sin 2θ = = . (x2 + y 2 )2 r4 r2 Similarly, x2 − y 2 r2 (cos2 θ − sin2 θ) cos 2θ = = . (x2 + y 2 )2 r4 r2 Since limr→∞ (sin 2θ)/r2 = limr→∞ (cos 2θ)/r2 = 0, V ≈ Ai for r large or (x, y) far from the origin. " ! 2Axy x2 − y 2 , Q(x, y) = − 2 , and R(x, y) = 0, we have (b) Identifying P (x, y) = A 1 − 2 (x − y 2 )2 (x + y 2 )2 Py = 2Ay(3x2 − y 2 ) , (x2 + y 2 )3 Qx = 2Ay(3x2 − y 2 ) , (x2 + y 2 )3 and Pz = Qz = Rx = Ry = 0. Thus, curl V = (Ry − Qz )i + (Pz − Rx )j + (Qx − Py )k = 0 and V is irrotational. (c) Since Px = 2Ax(x2 − 3y 2 ) 2Ax(3y 2 − x2 ) , Q = , and Rz = 0, ∇ · F = Px + Qy + Rz = 0 and V is y (x2 + y 2 )3 (x2 + y 2 )3 incompressible. 469 9.7 Divergence and Curl 44. We first note that curl (∂H/∂t) = ∂(curl H)/∂t and curl (∂E/∂t) = ∂(curl E)/∂t. Then, from Problem 36, and ∇2 E = 1 c2 −∇2 E = −∇2 E + 0 = −∇2 E + grad 0 = −∇2 E + grad (div E) = curl (curl E) * ) * ) 1 ∂ 1 ∂E 1 ∂ 1 ∂2E 1 ∂H curl H = − =− =− 2 = curl − c ∂t c ∂t c ∂t c ∂t c ∂t ∂ 2 E/∂t2 . Similarly, 2 ) 2 −∇ H = −∇ H + grad (div H) = curl (curl H) = curl = 1 ∂ c ∂t ) − 1 ∂H c ∂t * =− 1 ∂2H c2 ∂t2 1 ∂E c ∂t * = 1 ∂ curl E c ∂t 1 2 ∂ H/∂t2 . c2 45. We note that div F = 2xyz − 2xyz + 1 = 1 ̸= 0. If F = curl G, then div(curl G) = div F = 1. But, by Problem 30, for any vector field G, div(curl G) = 0. Thus, F cannot be the curl of G. and ∇2 H = EXERCISES 9.8 Line Integrals 1. √ * (π/4 ( 125 2 1 sin3 t (( = − 3 6 0 0 C 0 * (π/4 ) ' π/4 ' π/4 ' ( 1 cos2 t sin t dt = 250 − cos3 t (( 2(5 cos t)(5 sin t)(5 cos t) dt = 250 2xy dy = 3 0 0 C 0 . √ / √ 125 2 250 = 1− = (4 − 2) 3 4 6 ' ' π/4 ' π/4 + 2xy ds = 2(5 cos t)(5 sin t) 25 sin2 t + 25 cos2 t dt = 250 sin t cos t dt ' 2xy dx = C ' π/4 2(5 cos t)(5 sin t)(−5 sin t) dt = −250 ' π/4 sin2 t cos t dt = −250 0 ) 0 * (π/4 ( 125 1 2 sin t (( = = 250 2 2 0 ' ' 1 ' 1 2. (x3 + 2xy 2 + 2x) dx = [8t3 + 2(2t)(t4 ) + 2(2t)]2 dt = 2 (8t3 + 4t5 + 4t) dt ) C 0 0 * (1 ( 28 2 6 2 4 = 2 2t + t + 2t (( = 3 3 0 ' ' 1 ' 1 (8t4 + 4t6 + 4t2 ) dt [8t3 + 2(2t)(t4 ) + 2(2t)]2t dt = 2 (x3 + 2xy 2 + 2x) dy = ) 0 0 C * (1 ( ( = 736 =2 ( 105 0 ' ' 1 ' + 3 2 3 4 2 (x + 2xy + 2x) ds = [8t + 2(2t)(t ) + 2(2t)] 4 + 4t dt = 8 ) C 8 5 4 7 4 3 t + t + t 5 7 3 0 =8 ) 0 1 (1 + t2 )7/2 7 * (1 ( ( = 8 (27/2 − 1) ( 7 0 470 1 t(1 + t2 )5/2 dt 9.8 3. ' (3x2 + 6y 2 ) dx = C ' 0 [3x2 + 6(2x + 1)2 ] dx = ' −1 −1 = −(−9 + 12 − 6) = 3 ' ' 0 (3x2 + 6y 2 ) dy = [3x2 + 6(2x + 1)2 ]2 dx = 6 C (3x2 + 6y 2 ) ds = ' x2 dx = y3 ' 8 ' x2 dy = y3 ' 8 ' x2 ds = y3 ' 8 ' z dx = C C C 5. (0 ( (27x2 + 24x + 6) dx = (9x3 + 12x2 + 6x) ( −1 −1 ' C 4. 0 Line Integrals ' 0 √ √ [3x2 + 6(2x + 1)2 ] 1 + 4 dx = 3 5 −1 x2 8 dx = 2 27x /8 27 1 8 dx = 1 x2 8 x−1/3 dx = 27x2 /8 27 1 1 ' ' 56 27 8 −1/3 x 1 x2 + 8 1 + x−2/3 dx = 2 27x /8 27 ' π/2 ' π/2 t(− sin t) dt 8 ' (8 4 4 2/3 (( dx = x ( =3 9 1 −1/3 x 1 + 8 (1 + x2/3 )3/2 1 + x2/3 dx = 27 Integration by parts (8 ( ( = 8 (53/2 − 23/2 ) ( 27 1 0 C (π/2 ( = (t cos t − sin t) ( = −1 0 ' C z dy = t cos t dt Integration by parts 0 (π/2 π ( = (t sin t + cos t) ( = − 1 2 0 (π/2 ' ' π/2 π2 1 ( z dz = t dt = t2 (( = 2 8 C 0 0 ' π/2 + ' √ ' 2 2 t sin t + cos t + 1 dt = 2 z ds = C 6. 0 0 π/2 √ π2 2 t dt = 8 (1 * ' 8 8 9 (( 8 1 8 1 3 t dt = t (t2 )(2t)t2 dt = t ( = 3 3 27 27 C 0 0 0 (1 * ' ' 1 ) ' 1 1 3 2 ( 16 2 4 4xyz dy = t7 dt = t8 (( = t (t2 )(2t)2t dt = 3 3 0 3 3 C 0 0 (1 * ' 1 ' 1 ) ' 16 16 7 (( 16 1 3 t (t2 )(2t)2 dt = t6 dt = t ( = 4 4xyz dz = 3 3 21 21 0 0 C 0 * * (1 ) ' ' 1 ) ' 1 + 1 3 8 1 9 2 7 (( 8 200 6 2 2 4 2 4 4xyz ds = t (t )(2t) t + 4t + 4 dt = t + t ( = t (t + 2) dt = 3 3 0 3 9 7 189 C 0 0 ' 4xyz dx = ' 1 4 ) 7. Using x as the parameter, dy = dx and ' (2x + y) dx + xy dy = C ' 2 (2x + x + 3 + x2 + 3x) dx = −1 = ) * (2 ( 1 3 2 x + 3x + 3x (( = 21. 3 −1 471 ' 2 −1 (x2 + 6x + 3) dx 9.8 Line Integrals 8. Using x as the parameter, dy = 2x dx and ' ' 2 ' 2 (2x + x + 1) dx + (2x + y) dx + xy dy = C = 2 x(x + 1) 2x dx = ' 2 (2x4 + 3x2 + 2x + 1) dx −1 −1 (2 −1 ) 2 * ( 141 2 5 3 2 x + x + x + x (( = . 5 5 −1 9. From (−1, 2) to (2, 2) we use x as a parameter with y = 2 and dy = 0. From (2, 2) to (2, 5) we use y as a parameter with x = 2 and dx = 0. ' ' 5 ' 2 (2 (5 ( ( 2y dy = (x2 + 2x) ( (2x + 2) dx + (2x + y) dx + xy dy = + y 2 ( = 9 + 21 = 30 C 2 −1 2 −1 10. From (−1, 2) to (−1, 0) we use y as a parameter with x = −1 and dx = 0. From (−1, 0) to (2, 0) we use x as a parameter with y = dy = 0. From (2, 0) to (2, 5) we use y as a parameter with x = 2 and dx = 0. ' ' 5 ' 2 ' 0 (0 (2 (5 1 ( ( ( 2y dy = − y 2 ( + x2 ( 2x dx + (−1) y dy + (2x + y) dx + xy dy = + y2 ( 2 2 −1 0 0 −1 2 C = 2 + 3 + 25 = 30 11. Using x as a the parameter, dy = 2x dx. ' ' y dx + x dy = C 1 x2 dx + 0 x(2x) dx = 0 12. Using x as the parameter, dy = dx. ' ' y dx + x dy = 1 x dx + 1 ' 0 ' 1 x dx = 1 ' 0 0 0 C 1 ' (1 ( 3x2 dx = x3 ( = 1 0 (1 ( 2x dx = x2 ( = 1 0 13. From (0, 0) to (0, 1) we use y as a parameter with x = dx = 0. From (0, 1) to (1, 1) we use x as a parameter with y = 1 and dy = 0. ' 1 ' 1 dx = 1 y dx + x dy = 0 + C 0 14. From (0, 0) to (1, 0) we use x as a parameter with y = dy = 0. From (1, 0) to (1, 1) we use y as a parameter with x = 1 and dx = 0. ' ' 1 y dx + x dy = 0 + 1 dy = 1 C 15. ' (6x2 + 2y 2 ) dx + 4xy dy = C 0 9 1 (6t + 2t2 ) t−1/2 dt + 2 4 (9 ( = (2t3/2 + 2t5/2 ) ( = 460 ' ' 9 ' 9 √ 4 t t dt = (3t1/2 + 5t3/2 ) dt 4 4 4 16. ' 17. ' (−y 2 ) dx + xy dy = C ' 2 (−t6 ) 2 dt + ' 2 (2t)(t3 )3t2 dt = 2x3 y dx + (3x + y) dy = C ' 1 ) 4 9 1 y + y3 + y2 9 2 2(y 6 )y 2y dy + −1 = 2 4t6 dt = 0 0 0 ' 1 ' * (3y 2 + y) dy = −1 (1 ( ( ( −1 ' 1 −1 = 26 9 472 (2 4 7 (( 512 t ( = 7 7 0 (4y 8 + 3y 2 + y) dy 9.8 18. ' ' 4x dx + 2y dy = C 2 4(y 3 + 1)3y 2 dy + ' 2 ' 2y dy = 2 (12y 5 + 12y 2 + 2y) dy −1 −1 −1 Line Integrals (2 ( = (2y 6 + 4y 3 + y 2 ) ( = 165 −1 19. From (−2, 0) to (2, 0) we use x as a parameter with y = dy = 0. From (2, 0) to (−2, 0) we parameterize the semicircle as x = 2 cos θ and y = 2 sin θ for 0 ≤ θ ≤ π. 3 ' ' 2 2 2 2 x dx + ˇ (x + y ) dx − 2xy dy = C π 4(−2 sin θ dθ) − 0 −2 ' π 8 cos θ sin θ(2 cos θ dθ) 0 (2 ' π 1 3 (( = x ( (sin θ + 2 cos2 θ sin θ) dθ −8 3 0 −2 ) * (π ( 16 2 64 16 80 = − 8 − cos θ − cos3 θ (( = − =− 3 3 3 3 3 0 20. We start at (0, 0) and use x as a parameter. 3 ' ' 1 2 4 2 2 (x + x ) dx − 2 ˇ (x + y ) dx − 2xy dy = C 1 2 xx (2x dx) + 0 √ x x 0 (x2 + x) dx 1 0 0 ' ) * 1 −1/2 −2 x dx 2 1 ' 1 ' 0 ' 1 3 2 2 4 (−3x4 ) dx = − x5 x dx = (x − 3x ) dx + = 5 0 1 0 ' (1 ( ( = −3 ( 5 0 21. From (1, 1) to (−1, 1) and (−1, −1) to (1, −1) we use x as a parameter with y = 1 and y = −1, respectively, and dy = 0. From (−1, 1) to (−1, −1) and (1, −1) to (1, 1) we use y as a parameter with x = −1 and x = 1, respectively, and dx = 0. 3 ' 2 3 2 x y dx − xy dy = ˇ C −1 x2 (1) dx + ' −1 1 1 (−1 1 1 ( + y3 = x3 (( 3 3 1 −(−1)y 2 dy + (−1 ( 1 ( − x3 ( 3 1 (1 ( ( ( −1 ' 1 x2 (−1)3 dx + ' 1 −1 −1 (1 1 ( 8 − y 3 (( = − 3 3 −1 −(1)y 2 dy 22. From (2, 4) to (0, 4) we use x as a parameter with y = 4 and dy = 0. From (0, 4) to (0, 0) we use y as a parameter with x = dx = 0. From (0, 0) to (2, 4) we use y = 2x and dy = 2 dx. 3 ' 0 ' 0 ' 2 ' 2 2 3 2 2 2 3 x y dx − xy dy = x (64) dx − 0 dy + x (8x ) dx − x(4x2 )2 dx ˇ C 2 ' 3 23. ˇ (x2 − y 2 ) ds = C 0 2π 4 (0 64 3 (( 4 = x ( + x6 3 3 2 (2 ( ( − 2x4 ( 0 0 0 (2 ( ( = − 512 + 256 − 32 = − 352 ( 3 3 3 0 ' + (25 cos2 θ − 25 sin2 θ) 25 sin2 θ + 25 cos2 θ dθ = 125 0 2π (cos2 θ − sin2 θ) dθ (2π ( 125 cos 2θ dθ = = 125 sin 2θ (( = 0 2 0 0 ' π ' π ' π ' π 3 24. ˇ y dx − x dy = dt = −6π (sin2 t + cos2 t) dt = −6 2 cos t(3 cos t) dt = −6 3 sin t(−2 sin t) dt − C 0 0 0 0 ' Thus, y dx − x dy = 6π. ' 2π −C 473 9.8 Line Integrals 25. We parameterize the line segment from (0, 0, 0) to (2, 3, 4) by x = 2t, y = 3t, z = 4t for 0 ≤ t ≤ 1. We parameterize the line segment from (2, 3, 4) to (6, 8, 5) by x = 2 + 4t, y = 3 + 5t, z = 4 + t, 0 ≤ t ≤ 1. 3 ' 1 ' 1 ' 1 ' 1 (3 + 5t)(4 dt) 2t(4 dt) + 4t(3 dt) + 3t(2 dt) + ˇ y dx + z dy + x dz = C + ' 0 0 0 0 1 ' (4 + t)(5 dt) + 1 (2 + 4t) dt 0 0 * (1 ( 55 2 123 (55t + 34) dt = = t + 34t (( = 2 2 0 0 ) * * ' 2 ' 2) ' 2 ' 5 5 2 (3t) t (3t2 dt) + t dt t3 (3 dt) + y dx + z dy + x dz = 26. 4 2 0 0 0 C * ) * (2 ' 2) 15 4 15 2 3 4 3 5 5 3 (( 3 = 3t + t + t dt = t + t + t ( = 56 4 2 4 4 2 0 0 ' 1 ) 27. From (0, 0, 0) to (6, 0, 0) we use x as a parameter with y = dy = 0 and z = dz = 0. From (6, 0, 0) to (6, 0, 5) we use z as a parameter with x = 6 and dx = 0 and y = dy = 0. From (6, 0, 5) to (6, 8, 5) we use y as a parameter with x = 6 and dx = 0 and z = 5 and dz = 0. ' ' y dx + z dy + x dz = C 6 0+ 0 ' 5 6 dz + 0 ' 8 5 dy = 70 0 28. We parametrize the line segment from (0, 0, 0) to (6, 8, 0) by x = 6t, y = 8t, z = 0 for 0 ≤ t ≤ 1. From (6, 8, 0) to (6, 8, 5) we use z as a parameter with x = 6, dx = 0, and y = 8, dy = 0. ' ' 5 ' 1 (1 ( 6 dz = 24t2 ( + 30 = 54 8t(6 dt) + y dx + z dy + x dz = 0 0 0 C 29. F = e3t i − (e−4t )et j = e3t i − e−3t j; dr = (−2e−2t i + et j) dt; F · dr = (−2et − e−2t ) dt; * (ln 2 ) * ) ' ' ln 2 3 19 31 1 −2t (( t −2t t =− F · dr = =− − − (−2e − e ) dt = −2e + e ( 2 8 2 8 C 0 0 3 6 30. F = et i + tet j + t3 et k; dr = (i + 2tj + 3t2 k) dt; ) * (1 ' 1 ' 2 t3 1 t6 (( 13 t 2 t3 5 t6 t (e + 2t e + 3t e ) dt = e + e + e F · dr = ( = 6 (e − 1) 3 2 0 C 0 1 31. Using x as a parameter, r(x) = xi + ln xj. Then F = ln xi + xj, dr = (i + j) dx, and x ' e ' (e ( (ln x + 1) dx = (x ln x) ( = e. F · dr = W = 1 1 C 32. Let r1 = (−2 + 2t)i + (2 − 2t)j and r2 = 2ti + 3tj for 0 ≤ t ≤ 1. Then dr1 = 2i − 2j, dr2 = 2i + 3j, F1 = 2(−2 + 2t)(2 − 2t)i + 4(2 − 2t)2 j = (−8t2 + 16t − 8)i + (16t2 − 32t + 16)j, F2 = 2(2t)(3t)i + 4(3t)2 j = 12t2 i + 36t2 j, and W = ' = ' F1 · dr1 + C1 0 1 ' C2 F2 · dr2 = ' 0 1 (−16t2 + 32t − 16 − 32t2 + 64t − 32) dt + (1 ( (84t2 + 96t − 48) dt = (28t3 + 48t2 − 48t) ( = 28. 0 474 ' 0 1 (24t2 + 108t2 ) dt 9.8 Line Integrals 33. Let r1 = (1 + 2t)i + j, r2 = 3i + (1 + t)j, and r3 = (3 − 2t)i + (2 − t)j for 0 ≤ t ≤ 1. Then dr1 = 2i, dr2 = j, dr3 = −2i − j, F1 = (1 + 2t + 2)i + (6 − 2 − 4t)j = (3 + 2t)i + (4 − 4t)j, F2 = (3 + 2 + 2t)i + (6 + 6t − 6)j = (5 + 2t)i + 6tj, F3 = (3 − 2t + 4 − 2t)i + (12 − 6t − 6 + 4t)j = (7 − 4t)i + (6 − 2t)j, and W = ' F1 · dr1 + ' 1 (6 + 4t)dt + ' 0 3 4 5 2 ' 1 ' F3 · dr3 C3 1 6tdt + 1 ' 0 0 0 = F2 · dr2 + C2 C1 = ' (−14 + 8t − 6 + 2t)dt (1 ( (−14 + 20t) dt = (−14t + 10t2 ) ( = −4. 0 34. F = t i + t j + t k; dr = 3t i + 2tj + k; W = ' F · dr = C ' 3 5 5 5 (3t + 2t + t ) dt = 1 ' 3 1 35. r = 3 cos ti + 3 sin tj, 0 ≤ t ≤ 2π; dr = −3 sin ti + 3 cos tj; F = ai + bj; ' ' 2π (2π ( W = F · dr = (−3a sin t + 3b cos t) dt = (3a cos t + 3b sin t) ( = 0 C (3 ( 6t5 dt = t6 ( = 728 1 0 0 36. Let r = ti + tj + tk for 1 ≤ t ≤ 3. Then dr = i + j + k, and ct c c (ti + tj + tk) = √ (i + j + k) = √ (i + j + k), 3 |r| 3 3 t2 ( 3t2 )3 * * (3 ) ) ' 3 c c c 2c 1 (( 1 c 1 √ √ √ = dt = + 1 = √ . − − (1 + 1 + 1) dt = √ t (1 3 3 3 t2 3 1 t2 3 3 3 3 F= W = ' F · dr = 1 C 37. ' ' y 2 dx + xy dy = ' 1 3 (4t + 2)2 2 dt + 1 (2t + 1)(4t + 2)4 dt = ' 1 (64t2 + 64t + 16) dt 0 0 0 C1 ' * (1 ( 64 64 3 208 = t + 32t2 + 16t (( = + 32 + 16 = 3 3 3 0 ( √3 ' √3 ' √3 ' ' √3 ( 8 208 8 4t4 (2t) dt + 2t4 (4t) dt = 16t5 dt = t6 (( = 72 − = y 2 dx + xy dy = 3 3 3 1 1 1 C2 1 (e3 ' e3 ' ' e3 ' e3 ( 1 8 2 8 208 8 4(ln t)2 dt + y 2 dx + xy dy = 2(ln t)2 dt = (ln t)2 dt = (ln t)3 (( = (27 − 1) = t t t 3 3 3 e C3 e e e √ * (2 ) ' ' ' 2 2 √ √ √ 1 3 (( 16 5 t ( = t2 dt = 2 5 38. xy ds = t(2t) 1 + 4 dt = 2 5 3 3 0 C1 0 0 ' ' 2 ' 2 + + t(t2 ) 1 + 4t2 dt = xy ds = t3 1 + 4t2 dt u = 1 + 4t2 , du = 8t dt, t2 = 41 (u − 1) ) 0 C2 0 √ * (17 391 17 + 1 2 5/2 2 3/2 (( u − u = ( 5 3 120 1 1 1 √ ! " (3 ' 3 ' ' 3 √ 1 √ ( √ 16 5 (2t − 4)(4t − 8) 4 + 16 dt = 16 5 xy ds = (t − 2)2 dt = 16 5 (t − 2)3 (( = 3 3 2 C3 2 2 = ' 17 1 1 1 (u − 1)u1/2 du = 4 8 32 ' 17 (u3/2 − u1/2 ) du = 1 32 ) C1 and C3 are different parameterizations of the same curve, while C1 and C2 are different curves. 475 9.8 Line Integrals 39. Since v · v = v 2 , d dv dv dv d 2 v = (v · v) = v · + ·v =2 · v. Then dt dt dt dt dt ) * * ' b ' b ) ' b ' dv 1 d 2 dr ma · dt = m · v dt = m v dt F · dr = W = dt dt a dt a 2 a C (b 1 1 1 ( = m(v 2 ) ( = m[v(b)]2 − m[v(a)]2 . 2 2 2 a 40. We are given ρ = kx. Then ' π ' ' π ' π + 2 2 (1 + cos t) dt m= ρ ds = kx ds = k (1 + cos t) sin t + cos t dt = k 0 C 0 0 (π ( = k(t + sin t) ( = kπ. 0 41. From Problem 40, m = kπ and ds = dt. ) * (π ' π ' ' ( 1 Mx = (1 + cos t) sin t dt = k − cos t + sin2 t (( = 2k kxy ds = k yρ ds = 2 0 C C 0 ' π ' π ' ' (1 + 2 cos t + cos2 t) dt (1 + cos t)2 dt = k kx2 ds = k xρ ds = My = 0 0 C C * (π ) ( 1 1 3 = k t + 2 sin t + t + sin 2t (( = kπ 2 4 2 0 3kπ/2 3 2k 2 = ; ȳ = Mx /m = = . The center of mass is (3/2, 2/π). kπ 2 kπ π 42. On C1 , T = i and F · T = compT F ≈ 1. On C2 , T = −j and F · T = compT F ≈ 2. On C3 , T = −i and F · T = compT F ≈ 1.5. Using the fact that the lengths of C1 , C2 , and C3 are 4, 5, and 5, respectively, we have ' ' ' ' F · T ds ≈ 1(4) + 2(5) + 1.5(5) = 21.5 ft-lb. F · T ds + F · T ds + F · T ds = W = x̄ = My /m = C C3 C2 C1 EXERCISES 9.9 Independence of Path 1 1. (a) Py = 0 = Qx and the integral is independent of path. φx = x2 , φ = x3 + g(y), 3 ' (2,2) ((2,2) 1 1 1 1 16 ( φy = g ′ (y) = y 2 , g(y) = y 3 , φ = x3 + y 3 , x2 dx + y 2 dy = (x3 + y 3 ) ( = 3 3 3 3 3 (0,0) (0,0) ' 2 ' (2,2) 2 ((2 16 (x2 + x2 ) dx = x3 ( = x2 dx + y 2 dy = (b) Use y = x for 0 ≤ x ≤ 2. 3 3 0 0 (0,0) 2. (a) Py = 2x = Qx and the integral is independent of path. φx = 2xy, φ = x2 y + g(y), ' (2,4) ((2,4) ( φy = x2 + g ′ (y) = x2 , g(y) = 0, φ = x2 y, 2xy dx + x2 dy = x2 y ( = 16 − 1 = 15 (1,1) (1,1) (b) Use y = 3x − 2 for 1 ≤ x ≤ 2. ' 2 ' 2 ' (2,4) (2 ( (9x2 − 4x) dx = (3x3 − 2x2 ) ( = 15 [2x(3x − 2) + x2 (3)] dx = 2xy dx + x2 dy = (1,1) 1 1 476 1 9.9 3. (a) Py = 2 = Qx and the integral is independent of path. φx = x + 2y, φ = Independence of Path 1 2 x + 2xy + g(y), 2 1 1 1 φy = 2x + g ′ (y) = 2x − y, g(y) = − y 2 , φ = x2 + 2xy − y 2 , 2 2 2 * ((3,2) ) ' (3,2) ( 1 1 2 x + 2xy − y 2 (( = 14 (x + 2y) dx + (2x − y) dy = 2 2 (1,0) (1,0) (b) Use y = x − 1 for 1 ≤ x ≤ 3. ' (3,2) (1,0) (x + 2y) dx + (2x − y) dy = ' = ' 3 [x + 2(x − 1) + 2x − (x − 1) dx 1 3 (3 ( (4x − 1) dx = (2x2 − x) ( = 14 1 1 4. (a) Py = − cos x sin y = Qx and the integral is independent of path. φx = cos x cos y, φ = sin x cos y + g(y), φy = − sin x sin y + g ′ (y) = 1 − sin x sin y, g(y) = y, φ = sin x cos y + y, ' (π/2,0) ((π/2,0) ( =1 cos x cos y dx + (1 − sin x sin y) dy = (sin x cos y + y) ( (0,0) (0,0) (b) Use y = 0 for 0 ≤ x ≤ π/2. ' (π/2,0) (0,0) cos x cos y dx + (1 − sin x sin y) dy = ' 0 π/2 (π/2 ( =1 cos x dx = sin x ( 0 x x x 1 5. (a) Py = 1/y 2 = Qx and the integral is independent of path. φx = − , φ = − +g(y), φy = 2 +g ′ (x) = 2 , y y y y ((4,4) ' (4,4) 1 x x x ( g(y) = 0, φ = − , =3 − dx + 2 dy = (− ) (( y y y y (4,1) (4,1) (b) Use x = 4 for 1 ≤ y ≤ 4. ' (4,4) (4,1) 1 x − dx + 2 dy = y y 4 ' 1 (4 4 4 (( dy = − ( = 3 y2 y 1 6. (a) Py = −xy(x2 + y 2 )−3/2 = Qx and the integral is independent of path. φx = + x x2 + y2 , + + y y + g ′ (y) = + , g(y) = 0, φ = x2 + y 2 , x2 + y 2 + g(y), φy = + 2 2 2 2 x +y x +y ' (3,4) ( x dx + y dy + 2 ((3,4) + =4 = x + y2 ( (1,0) x2 + y 2 (1,0) φ= (b) Use y = 2x − 2 for 1 ≤ x ≤ 3. ' (3,4) (1,0) x dx + y dy + = x2 + y 2 ' 1 3 x + (2x − 2)2 + dx = x2 + (2x − 2)2 ' 1 3 √ (3 + 5x − 4 ( = 5x2 − 8x + 4 ( = 4 1 5x2 − 8x + 4 7. (a) Py = 4xy = Qx and the integral is independent of path. φx = 2y 2 x − 3, φ = x2 y 2 − 3x + g(y), φy = 2x2 y + g ′ (y) = 2x2 y + 4, g(y) = 4y, φ = x2 y 2 − 3x + 4y, ' (3,6) ((3,6) ( (2y 2 x − 3) dx + (2yx2 + 4) dy = (x2 y 2 − 3x + 4y) ( = 330 (1,2) (1,2) 477 9.9 Independence of Path (b) Use y = 2x for 1 ≤ x ≤ 3. ' ' (3,6) 2 2 (2y x − 3) dx + (2yx + 4) dy = 3 1 (1,2) = ' 3 1 2 [2(2x)2 x − 3] + [2(2x)x2 + 4]2 dx (3 ( (16x3 + 5) dx = (4x4 + 5x) ( = 330 1 1 8. (a) Py = 4 = Qx and the integral is independent of path. φx = 5x + 4y, φ = 5 2 x + 4xy − 2y 4 , 2 * ((0,0) ) ' (0,0) ( 5 2 7 4 3 (5x + 4y) dx + (4x − 8y ) dy = x + 4xy − 2y (( = 2 2 (−1,1) (−1,1) 5 2 x + 4xy + g(y), 2 φy = 4x + g ′ (y) = 4x − 8y 3 , g(y) = −2y 4 , φ = (b) Use y = −x for −1 ≤ x ≤ 0. ' ' (0,0) (5x + 4y) dx + (4x − 8y 3 ) dy = (−1,1) 0 −1 = ' [(5x − 4x) + (4x + 8x3 )(−1)] dx 0 −1 ) (−3x − 8x3 ) dx = 3 − x2 − 2x4 2 9. (a) Py = 3y 2 + 3x2 = Qx and the integral is independent of path. φx = y 3 + 3x2 y, * (0 ( ( ( = −1 7 2 φ = xy 3 + x3 y + g(y), φy = 3xy 2 + x3 + g ′ (y) = x3 + 3y 2 x + 1, g(y) = y, φ = xy 3 + x3 y + y, ' (2,8) ((2,8) ( (y 3 + 3x2 y) dx + (x3 + 3y 2 x + 1) dy = (xy 3 + x3 y + y) ( = 1096 (0,0) (0,0) (b) Use y = 4x for 0 ≤ x ≤ 2. ' (2,8) ' 2 (y 3 + 3x2 y) dx + (x3 + 3y 2 x + 1) dy = [(64x3 + 12x3 ) + (x3 + 48x3 + 1)(4)] dx (0,0) 0 = ' 0 2 (2 ( (272x3 + 4) dx = (68x4 + 4x) ( = 1096 0 10. (a) Py = −xy cos xy − sin xy − 20y 3 = Qx and the integral is independent of path. φx = 2x − y sin xy − 5y 4 , φ = x2 + cos xy − 5xy 4 + g(y), φy = −x sin xy − 20xy 3 + g ′ (y) = −20xy 3 − x sin xy, g(y) = 0, φ = x2 + cos xy − 5xy 4 , ' (1,0) ((1,0) ( = −3 (2x − y sin xy − 5y 4 ) dx − (20xy 3 + x sin xy) dy = (x2 + cos xy − 5xy 4 ) ( (−2,0) (−2,0) (b) Use y = 0 for −2 ≤ x ≤ 1. ' ' (1,0) (2x − y sin xy − 5y 4 ) dx − (20xy 3 + x sin xy) dy = (−2,0) 1 −2 (1 ( 2x dx = x2 ( = −3 −2 11. Py = 12x3 y 2 = Qx and the vector field is a gradient field. φx = 4x3 y 3 + 3, φ = x4 y 3 + 3x + g(y), φy = 3x4 y 2 + g ′ (y) = 3x4 y 2 + 1, g(y) = y, φ = x4 y 3 + 3x + y 12. Py = 6xy 2 = Qx and the vector field is a gradient field. φx = 2xy 3 , φ = x2 y 3 + g(y), φy = 3x2 y 2 + g ′ (y) = 3x2 y 2 + 3y 2 , g(y) = y 3 , φ = x2 y 3 + y 3 13. Py = −2xy 3 sin xy 2 + 2y cos xy 2 , Qx = −2xy 3 cos xy 2 − 2y sin xy 2 and the vector field is not a gradient field. 14. Py = −4xy(x2 + y 2 + 1)−3 = Qx and the vector field is a gradient field. 478 9.9 Independence of Path 1 φx = x(x2 + y 2 + 1)−2 , φ = − (x2 + y 2 + 1)−1 + g(y), φy = y(x2 + y 2 + 1)−2 + g ′ (y) = y(x2 + y 2 + 1)−2 , 2 1 g(y) = 0, φ = − (x2 + y 2 + 1)−1 2 1 15. Py = 1 = Qx and the vector field is a gradient field. φx = x3 + y, φ = x4 + xy + g(y), φy = x + g ′ (y) = x + y 3 , 4 1 4 1 4 1 4 g(y) − y , φ = x + xy + y 4 4 4 16. Py = 4e2y , Qx = e2y and the vector field is not a gradient field. ' −y F · dr is independent of the path. Thus, instead of the given 17. Since Py = −e = Qx , F is conservative and C curve we may use the simpler curve C1 : y = x, 0 ≤ x ≤ 1. Then ' (2x + e−y ) dx + (4y − xe−y ) dy W = C1 = ' 1 (2x + e−x ) dx + 0 ' 1 (4x − xe−x ) dx 0 Integration by parts (1 (1 ( ( = (x2 − e−x ) ( + (2x2 + xe−x + e−x ) ( 0 0 = [(1 − e−1 ) − (−1)] + [(2 + e−1 + e−1 ) − (1)] = 3 + e−1 . 18. Since Py = −e−y = Qx , F is conservative and ' F · dr is independent of the path. Thus, instead of the given C curve we may use the simpler curve C1 : y = 0, −2 ≤ −x ≤ 2. Then dy = 0 and ' ' −2 (−2 ( −y −y W = (2x + e ) dx + (4y − xe ) dy = (2x + 1) dx = (x2 + x) ( = (4 − 2) − (4 + 2) = −4. C1 2 2 19. Py = z = Qx , Qz = x = Ry , Rx = y = Pz , and the integral is independent of path. Parameterize the line segment between the points by x = 1 + t, y = 1 + 3t, z = 1 + 7t, for 0 ≤ t ≤ 1. Then dx = dt, dy = 3 dt, dz = 7 dt and ' (2,4,8) ' 1 yz dx + xz dy + xy dz = [(1 + 3t)(1 + 7t) + (1 + t)(1 + 7t)(3) + (1 + t)(1 + 3t)(7)] dt (1,1,1) 0 = ' 0 1 (1 ( (11 + 62t + 63t2 ) dt = (11t + 31t2 + 21t3 ) ( = 63. 0 20. Py = 0 = Qx , Qz = 0 = Ry , Rx = 0 = Pz and the integral is independent of path. Parameterize the line segment between the points by x = t, y = t, z = t, for 0 ≤ t ≤ 1. Then dx = dy = dz = dt and ' (1,1,1) ' 1 (1 ( (2t + 3t2 + 4t3 ) dt = (t2 + t3 + t4 ) ( = 3. 2x dx + 3y 2 dy + 4z 3 dz = (0,0,0) 0 0 21. Py = 2x cos y = Qx , Qz = 0 = Ry , Rx = 3e3z = Pz , and the integral is independent of path. Integrating φx = 2x sin y + e3z we find φ = x2 sin y + xe3z + g(y, z). Then φy = x2 cos y + gy = Q = x2 cos y, so gy = 0, g(y, z) = h(z), and φ = x2 sin y + xe3z + h(z). Now φz = 3xe3z + h′ (z) = R = 3xe3z + 5, so h′ (z) = 5 and h(z) = 5z. Thus φ = x2 sin y + xe3z + 5z and ' (2,π/2,1) (2x sin y + e3z ) dx + x2 cos y dy + (3xe3z + 5) dz (1,0,0) ((2,π/2,1) ( = (x2 sin y + xe3z + 5z) ( = [4(1) + 2e3 + 5] − [0 + 1 + 0] = 8 + 2e3 . (1,0,0) 479 9.9 Independence of Path 22. Py = 0 = Qx , Qz = 0 = Ry , Rx = 0 = Pz , and the integral is independent of path. Parameterize the line segment between the points by x = 1 + 2t, y = 2 + 2t, z = 1, for 0 ≤ t ≤ 1. Then dx = 2 dt, dz = 0 and ' 1 ' (3,4,1) 1 [(2 + 4t + 1)2 + 3(2 + 2t)2 2] dt (2x + 1) dx + 3y 2 dy + dz = z 0 (1,2,1) ' 1 (1 ( (24t2 + 56t + 30) dt = (8t3 + 28t2 + 30t) ( = 66. = 0 0 23. Py = 0 = Qx , Qz = 0 = Ry , Rx = 2e2z = Pz and the integral is independent of path. Parameterize the line segment between the points by x = 1 + t, y = 1 + t, z = ln 3, for 0 ≤ t ≤ 1. Then dx = dy = dt, dz = 0 and ' 1 ' (2,2 ln 3) (1 ( [e2 ln 3 + 3(1 + t)2 ] dt = [9t + (1 + t)3 ] ( = 16. e2z dx + 3y 2 dy + 2xe2z dz = 0 0 (1,1,ln 3) 24. Py = 0 = Qx , Qz = 2y = Ry , Rx = 2x = Pz and the integral is independent of path. Parameterize the line segment between the points by x = −2(1 − t), y = 3(1 − t), z = 1 − t, for 0 ≤ t ≤ 1. Then dx = 2 dt, dy = −3 dt, dz = −dt, and ' (0,0,0) 2xz dx + 2yz dy + (x2 + y 2 ) dz (−2,3,1) = ' 1 [−4(1 − t)2 (2) + 6(1 − t)2 (−3) + 4(1 − t)2 (−1) + 9(1 − t)2 (−1)] dt 0 = ' 0 1 (1 ( −39(1 − t)2 dt = 13(1 − t)3 ( = −13. 0 25. Py = 1−z sin x = Qx , Qz = cos x = Ry , Rx = −y sin x = Pz and the integral is independent of path. Integrating φx = y − yz sin x we find φ = xy + yz cos x + g(y, z). Then φy = x + z cos x + gy (y, z) = Q = x + z cos x, so gy = 0, g(y, z) = h(z), and φ = xy + yz cos x + h(z). Now φz = y cos x + h(z) = R = y cos x, so h(z) = 0 and φ = xy + yz cos x. Since r(0) = 4j and r(π/2) = πi + j + 4k, ((π,1,4) ' ( F · dr = (xy + yz cos x) (( = (π − 4) − (0 + 0) = π − 4. C (0,4,0) 26. Py = 0 = Qx , Qz = 0 = Ry Rx = −ez = Pz and the integral is independent of path. Integrating φx = 2 − ez we find φ = 2x − xez + g(y, z). Then φy = gy = 2y − 1, so g(y, z) = y 2 − y + h(z) and φ = 2x − xez + y 2 − y + h(z). Now φz = −xez + h′ (z) = R = 2 − xez , so h′ (z) = 2, h(z) = 2z, and φ = 2x − xez + y 2 − y + 2z. Thus ((2,4,8) ' ( z 2 F · dr = (2x − xe + y − y + 2z) (( = (4 − 2e4 + 16 − 4 + 16) − (−2 + e−1 + 1 − 1 − 2) = 36 − 2e4 − e−1 . C (−1,1,−1) 27. Since Py = Gm1 m2 (2xy/|r|5 ) = Qx , Qz = Gm1 m2 (2yz/|r|5 ) = Ry , and Rx = Gm1 m2 (2xz/|r|5 ) = Pz , the force field is conservative. x , φ = Gm1 m2 (x2 + y 2 + z 2 )−1/2 + g(y, z), φx = −Gm1 m2 2 2 (x + y + z 2 )3/2 y y φy = −Gm1 m2 2 + gy (y, z) = −Gm1 m2 2 , g(y, z) = h(z), (x + y 2 + z 2 )3/2 (x + y 2 + z 2 )3/2 φ = Gm1 m2 (x2 + y 2 + z 2 )−1/2 + h(z), z z + h′ (z) = −Gm1 m2 2 , φz = −Gm1 m2 2 2 2 3/2 2 (x + y + z ) (x + y + z 2 )3/2 Gm1 m2 Gm1 m2 h(z) = 0, φ = + = 2 2 2 |r| x +y +z 480 9.10 Double Integrals 28. Since Py = 24xy 2 z = Qx , Qz = 12x2 y 2 = Ry , and Rx = 8xy 3 = Pz , F is conservative. Thus, the work done between two points is independent of the path. From φx = 8xy 3 z we obtain φ = 4x2 y 3 z which is a potential function for F. Then ' (1,√3 ,π/3) ' (0,2,π/2) ((1,√3 ,π/3) √ 2 3 ( F · dr = 4x y z ( W = F · dr = 0. = 4 3 π and W = (2,0,0) (2,0,0) 29. Since F is conservative, ' F · dr = C1 C is composed of C1 and C2 , 3 ' ˇ F · dr = C ' (2,0,0) F · dr. Then, since the simply closed curve −C2 F · dr + C1 ' F · dr = C2 ' C1 F · dr − ' F · dr = 0. −C2 30. From F = (x2 + y 2 )n/2 (xi + yj) we obtain Py = nxy(x2 + y 2 )n/2−1 = Qx , so that F is conservative. From φx = x(x2 + y 2 )n/2 we