UNIT Introduction to Differential Equation In this unit, important terms and concepts related to the solutions of differential equations that will be used throughout this course. This introduction will guide the student as he/she will proceeds to the next unit. 2 Essential Questions 1. How differential equation is formed? 2. What are the solution/methods in eliminating the arbitrary constantsof differential equations? 3. How differential equation can be used to trace families of curves? Intended Learning Outcomes 1. Define and distinguish the types, order, degree, and solutions of differential equations. 2. Explain the formulation of differential equation and exemplify methods of eliminating the arbitrary constants. 3. Solve problems involving families of curves in context such as, but not limited to, straight lines, circles and parabolas. Diagnostic Assessment #2 State the order and degree of each differental equation. e.g. 2nd order, 2nd degree 1. y = 2x 2. + %. & %' - %& %' %& - + x)1 + + , , ++ . %' %/& %' 0 , =4 / % 2& %'2 3. √1 − x - dx + 71 − y - dy = 0 2& - 0 4. +% 2, + +%&, = 4x + y %' %' _____ order,______degree _____ order,______degree _____ order,______degree _____ order,______degree Select the correct answer for each question. 1. Find the diferential equation of the family of lines passing through the origin. A. xdx – ydy = 0 C. xdx + ydy = 0 B. ydx – xdy = 0 D. ydx + xdy = 0 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation 2. Determine the differential equation of the family of lines passing through (h, k). A. (y – k)dx – (x – h)dy = 0 C. (x – h)dx – (y – k)dy = 0 B. (x – h) + (y – k) = dy/dx D. (x – h)dx + (y – k)dy = dy – dx 3. Determine the differential equation of the family of circles with center on the y-axis. A. (y'')3 - xy'' + y' = 0 C. xy'' – (y')3 – y' = 0 B. y'' – xyy' = 0 D. (y')3 + (y'')2 + xy = 0 4. Find the differential equation of the family of circles having center on y-axis and a radius of 3. A. (x2 – 9)(y')2 + x2 = 0 C. (x2 + 9)(y')2 – x2 = 0 B. (x2 – 9)(y')2 + y2 = 0 D. (x2 + 9)(y')2 – y2 = 0 5. Find the differential equation whose general solution is y = – C1x + C2ex. A. xy' – y''(x – 1) + y = 0 C. xy' – y''(x – 1) – y = 0 B. – xy' + y''(x – 1) + y= 0 D. – xy' – y''(x – 1) – y = 0 6. Find the differential equaton having a solution of y = C1 cos 2x + C2 sin 2x. A. y'' – 4y = 0 C. y'' – 4y' + 4y = 0 B. y'' + 4y= 0 D. y'' + 4y' + 4y = 0 7. The equation y2 = C/x is the general solution of what differential equation? A. y' = 2xy B. y' = – 2xy C. y' = – y / 2x D. y' = 2x / y 8. The equation 3xy = Ce – 2x is the general solution of A. xy' + x(2y + 1) = 0 C. yy' – x(2y + 1) = 0 B. yy' – y(2x + 1)= 0 D. xy' + y(2x + 1) = 0 9. Find the differential equation of the family of circles touching the y-axis at the origin. A. 2xydy – (x2 – y2)dx = 0 C. ydy – (x2 + y2)dx = 0 B. 2xydx + (x2 + y2)dy = 0 D. ydx + (x2 – y2)dy = 0 10. Find the differential equation of the family of circles with center at (0, 0). A. xy' –y = 0 B. xy' + y = 0 C. yy' + x = 0 D. yy' – x = 0 11. y = x2 + x + C is the solution of what differential equation? A. y' = 2x + 1 C. xy' = x2 + y B. y' = x2 + 2x D. xy' = 2x + 1 12. Which of the following equation has a solution of y = x2 + Cx? A. y' = 2x + 1 C. xy' = x2 + y B. y' = x2 + 2x D. xy' = 2x + 1 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation 2 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation 2.1. Definitions and Notations of Differential Equations Differential Equation is an equation containing derivatives or differentials. a. Ordinary differential equation involves only one independent variable 2 x − yy9 = 0 y" = − ; r %= + 2h = 0 &2 %> 2 % & %'2 = −k -y b. Partial differential equation contains partialderivatives A 2 B A 2B + =0 x AB + y AB = 2u A'2 A&2 A' A& Order of the Differential Equation – order of the highest derivative that occurs in it. Degree of the Differential Equation – its algebraic degree in the highest ordered derivative present. /& %& Example: % + %' %'/ - + y = x- 3rd order, 1st degree Example 2.1 Obtain the order and the degree (if possible) of the following differential equations. 1. 2 % & / %'2 = )1 + +%&, - %' FG 2. xe + %& FH +2= 0 4. + %'02 2 % & %'2 %' Answers: 1. 2nd order, 3rd degree 2. 1st order, undefined degree D 2& 3. x % + +%&, + xy = 0 %' = sin y + 3x , 3. 2nd order, 1st degree 4. 2nd order, 3rd degree EXERCISE #2 Obtain the order and the degree (if possible) of the following differential equations. %& - 1. x + + , %' 2. 2 % & %'2 % 3. ) / = 71 + y + sin +%&, + y = 0 %' 2& %'2 4. %& %' ' + FG = 0 FH 2& 0 5. x +% %'2 M , + y +%& ,− 5y = 0 %' %& =) %' 3 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation Solution of the Differential Equation – any relation free from derivatives, which involve one or more of the variables, and which is consistent with the equation. (“n”– th order corresponds to “n” arbitrary constants) 2' The equation % + k - x = 0 has the solution x = A cos kt + B sin kt %O2 a. General Solution – a solution involving “n” distinct arbitrary constants in a differential equation of order “n”. b. Particular Solution – any solution with no arbitrary constants since these are determined from initial value conditions. Example: The general solution of the equation xy’ – 2y = 0 is y = Cx-. If the graph passes through the point (1, 3), the particular solution is y = 3x-. 2.2. Formulation of Differential Equations Eliminate the arbitrary constants of the general solution by differentiation and algebraic substitution. Example 2.2 Find the differential equation for each of the following solution. a. y = Cx3 b. y = C1ex + C2e – x c. 2y2 – x2 = C d. y = C1e2x + C2xe2x e. y = ex(C1 cos x + C2 sin x) Solution: a. Differentiate y = Cx3 V y′ = C(3x-), then, C = & Therefore, 0'2 V y = + & 2,x 0 0' 3y = xy′ WX 9 − YZ = [ b. y = C\ e' + C- e]' y 9 = C\ e' − C- e ]' y 99 = C\ e' + C- e]' Therefore, y99 = y X99 − X = [ 4 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation c. 2y - − x - = C 4yy 9 − 2x = 0 ^XX 9 − W = [ d. y = C\ e-' + C- xe-' y 9 = 2C\ e-' + C- (e-' + 2xe-' ) y′ = 2C\ e-' + 2C- xe-' + C- e-' y′ = 2y + C- e-' y 99 = 2y 9 + 2C- e-' But C- e-' = y 9 − 2y, then y′′ = 2y9 + 2(y9 − 2y) y99 = 4y9 − 4y X99 − _X9 + _X = [ e. y = e' (C\ cos x + C- sin x) y 9 = e' (C\ cos x + C- sin x) + e' (−C\ sin x + C- cos x) y 9 = y + e' (−C\ sin x + C- cos x) y 99 = y 9 + e' (−C\ sin x + C- cos x) − e' (C\ cos x + C- sin x) But y 9 − y = e' (−C\ sin x + C- cos x) and y = e' (C\ cos x + C- sin x), then y99 = y9 + (y9 − y) − y X99 − ^X9 + ^ = [ A given function is said to be a particular solution of “n”-th order differential equation if the derivatives of the function up to “n”-th derivative will be substitute in the equation and makes the equation correct. Example 2.3 Verify that the given function y = e – 3x is a particular solution of the equation y'' + y' – 6y = 0. Solution: y' = – 3e – 3x and y'' = 9e – 3x y'' + y' – 6y = (9e – 3x) + (– 3e – 3x) – 6(e – 3x) = 9e – 3x – 3e – 3x – 6e – 3x y'' + y' – 6y = 0 Then y = e – 3x is a particular solution of y'' + y' – 6y = 0. 5 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation EXERCISE #3 Verify that the given function is a solution of the given differential equation. 1. y = x- + 4x ; xy9 = x- + y 2. y = (x + 1)e]' ; y 9 + y − e ]' = 0 3. y = e' cos x ; y 99 − 2y 9 + 2y = 0 2.3. Family of Curves Elimination of the arbitrary constants from the standard forms of the equation of a curve such as line, circle, parabola, ellipse, hyperbola, and other curves. Example 2.4 Find the differential equation of the concentric circles whose center is at (0, 0). Solution: For radius (r) is constant and (h, k) is at (0,0), x - + y- = r 2x dx + 2y dy = 0 x dx + y dy = 0 c W+ X = [ cW Figure 2.1 Example 2.5 Find the differential equation of the parabolas having vertex at origin whose axis is parallel to x-axis and opens to the right. Solution: y - = 4ax 2y dy = 4a dx 2yy′ = 4a Then, y - = (2yy 9 ) x y = 2x %& %' ^WcX − XcW = [ Figure 2.2 6 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation Example 2.6 Find the differential equation of the parallel lines represented by y = 2x + C. Solution: y = 2x + C dy = 2 dx cX − ^ =[ cW Example 2.7 Find the differential equation of the curve y = a sin (x + b). Solution: Let a and b be the arbitrary constants, y = a sin (x + b) y' = a cos (x + b) y'' = d – a sin (x + b) Then, y'' = – y c ^X cW ^ +X=[ Example 2.8 Find the differential equation of the ellipses having foci on x-axis and center at the origin. Solution: x- y + =1 a - bb- x - + a - y - = a - b2b- x + 2a- yy 9 = 0 byy9 =− a x x[yy 99 + (y 9 )- ] − yy′ =0 xWXX 99 + W(X 9 )^ − XX 9 = [ Figure 2.3 7 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation Example 2.9 Find the differential equation of the circles having center on the x-axis and radius of 1. Solution: At center (h, 0) and r = 1, (x − h)- + y- = 1 2(x − h) + 2yy9 = 0 x − h + yy9 = 0 h = x + yy9 Substitute h in the standard form ofequation [x − (x + yy 9 )]- + y - = 1 y - (y 9 )- + y - = 1 X ^ [(X 9 )^ + h] = h Figure 2.4 EXERCISE #3 Find the differential equation of the following family of curves 1. Hyperbolas having foci on x-axis and center at origin 2. Circles having center on y-axis and radius of 3 3. Lines passing through the point (h, k) 4. Circles touching the y-axis at the origin 5. Circles which touch the x and y axes in the first quadrant REFERENCES Dela Fuente, R. A., Uy, F. B., Templo Jr., P. T., & Ocampo, J. L.(2014). Simplified Differential Equations (3rd Ed.) Chapter 1, pp. 1 to 8. Merriam & Webster Bookstore, Inc. Rainville, E. D., Bedient, P. E., & Bedient, R. E. (2001). Elementary Differential Equations – International Ed. (8th Ed.) Chapter 1, pp. 1 to 17. Pearson Education Asia Pte. Ltd. Zill D. G. & Cullen, M. R. (2013). Differential Equations with Boundary-Value Problems (7th ed) Chapter 1 pp. 2 to 9, 13 to 16. Cengage Learning 8 DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation Find the order and degree of each of the following differential equations. %2 i %i 0 1. y = x %& + \ 5. + %O2 , + 3 + , + 4 = 0 FG %' %O FH 2. +% .& - %'. 3. %O %' 4. %/& 0 , + +%'/, = 4 %2& %' 2 6. %2 i +%O , + 3s %O2 = 0 y′′ )- + y′ = 3x 7. √1 − x-dx + 71 − y - dy = 0 + 5xt = 0 %i D ( 8. ( y 999 )- 9 ( )D + y′′ =y+y−5 Find the differential equations of the following solutions. 1. x2 + 2xy = C 4. y = C1ex + C2 xex 2. y = C1 ex + C2 e –3x 5. ye2x – 2cos x = C 3. y = C1 sin 2x + C2 cos 2x 6. y = C1 e2x + C2 e – x Find the differential equations described by the following families of curves. 1. y2 = m(a2 – x2) 2. Circles touching x-axis at the origin 3. Vertical parabolas whose vertex is at the origin 4. Straight lines with slope and y-intercept equal 5. Circles which pass through the origin and whose centers lie on y-axis 6. Horizontal parabolas whose vertex is at the origin 7. Straight lines passing through the point (3, 2) 8. Circles with center on the y-axis DIFFERENTIAL EQUATION: A SIMPLIFIED APPROACH Introduction to Differential Equation 9