Math 220, Differential Equations • Professor Charles S.C. Lin • Office: 528 SEO, Phone: 413-3741 • Office Hours: MWF 2:00 p.m. & by appointments • E-mail address: cslin@uic.edu 1 Teaching Assistant • • • • • Mr. Diego Dominici Office: 607 SEO Office Hours: ? Phone: 996-4814 e-mail: ddomin1@uic.edu 2 Please check: • www.math.uic.edu/~berger/M220/index.ht ml for Syllabus, assignments, etc... • www.awlonline.com/nagle for interactive CD 3 Differential Equation: Classifications • Ordinary differential equations, order! • Partial differential equations • Linear equations: i.e. linear in the dependent variable(s). • Nonlinear differential equations: not linear • For example: xy" y ' xy 0 This eq. appears in stress analysis, aerodynami cs. 4 Explicit solution and Implicit solution • If a function satisfies a differential equation, for example: 3 y x 8 , satisfy th e D.E. dy 3 x 2 which can be verified easily. dx 2 y • Such a function, defined explicitly as a function of independent variable x is called an explicit solution. On the other hand, the equation x y e xy 0, satisfies the D.E. 5 Given by • the following: (*) • The equation dy (1 ye xy ) . xy dx (1 xe ) x ye 0 xy • is said to defined an implicit solution of the equation (*) above. 6 In fact, there are many solutions to a D.E such as (*) above. • To find a solution passing through a specific point in xy-plane, we need to impose a condition, known as : initial value, i.e. y(x0) = y0. This is known as the initial value problem. We shall assume that the function f(x,y) is sufficiently smooth, that a solution always exists. Namely: • Theorem (Existence and uniqueness): The I.V.P. y' f ( x, y) , y( x0 ) y0 • always has a unique solution in a rectangle containing the point (x0, y0), if f and f x are continuous there. 7 Direction Fields • Consider the first order D. E. dy f ( x, y) dx • the equation specifies a slope at each point in the xy-plane where f is defined. • It gives the direction that a solution to the equation must have at each point. 8 A plot of short line segments drawn at various points in the xyplane showing the slope of the solution curve this is called a “ direction field ” for the differential equation. • The direction field gives us the “ flow of solutions ”. 9 Example • For the equation dy x 2 y. dx • • • • Using Maple with(DEtools); eq:=diff(y(t),t)=t^2-y(t); DEplot(eq,y(t),t=-5..5,y=-5..5,arrows=slim) 10 Using Maple program, we have the following • graph: 11 Another example: • Consider the logistic equation for the population of a certain species: dp 3 p 2 p 2 , with p(0) 2 . dt • • • • using maple, we write in commands: eq:=diff(p(t),t)=3*p(t)-2*p(t)^2; DEplot(eq,p(t), t=0..5,p=0..5,arrows=slim); and get 12 Its direction field • like this: 13 The Method of Isoclines • Consider the differential equation • (*) dy/dx = f(x,y). • The set of points in the xy-plane where all the solutions have the same slope dy/dx; i.e. the level curves for the function f(x,y) are called the isoclines for the D. E. (*). • This is the family of curves f(x,y) = C. • This gives us a way to draw direction field. 14 Example • For the differential equation • • • • y' x y f(x,y) = x + y, and the set of points where: x + y = c, are straight lines with slope (-1). We can now draw the isoclines for the D. E. and the solution passing through a given initial point can also be drawn. 15 Let us graph the isoclines of f(x,y) = x + y. • and compare it to the direction field of it, we see 16 Maple example • Let us consider the IVP for y’ = x^2 - y, with three sets of initial points: [0,-1], [0,0] and [0,2]. What will be the corresponding solutions? 17 Separable Equation • Given a differential equation dy f ( x, y) dx • If the function f(x,y) can be written as a product of two functions g(x) and h(y), i.e. • f(x,y) = g(x) h(y), then the differential eq. is called separable. 18 Example • The equation dy 3x xy 2 dx 1 y • is separable, since 3 y 3x xy 2 ( x ) 2 g ( x ) h( y ) 1 y 1 y 19 Method for solving separable equation • Separable equation can be solved easily, • Rewrite the equation: dy g ( x ) h ( y ) in the form dx dy g ( x ) dx then find h( y ) antiderivatives on both sides. i. e. dy h( y ) g ( x ) dx C. 20 Example • Consider the initial value problem dy y 1 dx x3 y ( 1) 0 21 Using Maple: • we can solve the IVP with the following Maple commands. • ODE:=diff(y(x),x)=(y(x)-1)/(x-3); • IC:=y(-1)=0; • IVP:={ODE,IC}; • GSOLN:=dsolve(ODE,y(x)); • Then use the IC to find the arbitrary constant. 22 Linear Equations • We shall study how one can solve a first order linear differential equation of the form: dy a1 ( x ) a0 ( x ) y b( x ), dx • We first rewrite the above equation in the so called “standard form”: y ' P( x ) y Q( x ). 23 Integration Factor • Suppose we multiply a function (x) to the above equation, we get: ( x) y' ( x) P( x) y ( x)Q( x) • Is it possible for us to find (x) such that the left hand side • ? d LHS [ ( x) y] dx 24 • Since d dy [ ( x ) y ] ( x ) '( x ) y dx dx • We see that this can be done, if ' ( x ) ( x ) P( x ), which implies d P( x )dx , by integration, we get ln| ( x )| P( x )dx. Hence ( x) e P ( x ) dx . 25 In this case, • we can solve it by integration. • Note that: [ ( x ) y ]' ( x )Q( x ), implies ( x ) y ( x )Q( x )dx c, consequently we have 1 y( x) ( x) ( x)Q( x)dx c. 26 Examples: • Consider the D.E. y ' y e . 3x • Solution: • Another example: solve the following initial value problem: dx 3 2 t 3t x t , x (2) 0. dt 27 Application: Mixing Problems (Compartmental Analysis) • Consider a large tank holding 1000 L of water into which a brine solution of salt begins to flow at a constant rate of 6 L/min. The solution inside the tank is kept well stirred and is flowing out of the tank at a rate of 6 L/min. If the concentration of salt in the brine entering the tank is 1 kg/L, determine when the concentration of salt in the tank will reach 0.5 kg/L 28 • Let x(t) be the mass of salt in the tank at time t. The rate at which salt enters the tank is equal to “input rate - output rate”. Thus dx x(t ) (6 l / min)( 1kg / l ) (6 l / min)( kg / l ) dt 1000 or we have : dx 3x 6 , with x(0) 0 dt 500 29 The equation is separable • We can solve it easily, using the initial condition, we get x (t ) 1000(1 e 3t 500 ). Thus, the concentration of salt in the tank at time t is: 3t x (t ) 1 500 1 e kg / L. When will this = ? 1000 2 3t 1 500 Set 1 e , we find t = 115.52 min. 2 30 Existence and Uniqueness Theorem • Suppose P(x) and Q(x) are continuous on the interval (a,b) that contains the point x0. Then the initial value problem: • y + P(x)y = Q(x), y(x0)=y0 • for any given y0. • has a unique solution on (a,b). 31 Application to Population Growth • If we assume that the growth rate of a population is proportional to the population present, then it leads to a D.E.: • Let p(t) be the population at time t. Let k > 0 be the proportionality constant for the growth rate and let p0 be the population at time t = 0. Then a • mathematical model for a population could be: dp kp, dt p( 0) p0 32 This can be solved easily. • Example: In 1790 the population of the United States was 3.93 million, and in 1890 it was 62.95 million. Estimate the U.S. population as a function of time. 33 Application to Newtonian Mechanics • The study of motion of objects and the effect of forces acting on those objects is called Mechanics. A model for Newtonian mechanics is based on Newton’s laws of motion: Let us consider an example: An object of mass m is given an initial velocity of v0 and allowed to fall under the influence of gravity. Assuming the gravitational force is constant and the force due to air resistance is proportional to the velocity of the object . Determine the equation of motion for this object. 34 Solution • Since the total force acting on the object is • F = FG - FA = mg - k v(t). And according to Newton’s 2nd law of motion, F = m a, we see that • m a = mg - k v. • Let x(t) be the position function of the object at time t, and • v(t) = dx/dt, a = dv/dt. 35 Equation of motion can be rewritten as: • The following separable initial value problem. dv k g v, dt m v (0) v0 , where • We can solve the equation easily, and obtain: mg mg v(t ) (v0 )e k k kt m . 36 Now, to find the position function x(t) • Suppose that at t = 0, the object is x0 units above the ground, i.e. x(0) = x0 . Then for the position function x(t), we have the following I.V.P. dx mg mg (v0 )e dt k k kt m , with x(0) x0 . • This can be solved easily. 37 We obtain: • The equation of motion: kt m mgt m mg x(t ) (v0 )(1 e ) x0 . k k k 38 Linear Differential Operators (4.2) • We shall now consider linear 2nd order equations of the form: 2 d y dy (*) a2 ( x) 2 a1 ( x) a0 ( x) y b( x), dx dx Where ai ( x), and b( x) are continuous in x on some interval I. We generally assume that a2 ( x) is never zero on I, thus (*) can be put in the form d2y dy () p ( x) q ( x) y g ( x), 2 dx dx The so called " Standard form". 39 Homogeneous equation associated with () • is the equation 2 d y dy p ( x) q ( x) y 0, 2 dx dx in the sence of " linear algebra". For simplicity we shall define L[y] LHS of this equation, i.e. L[y] : y" p(x)y' q(x)y . L is called a differenti al operator of order 2. 40 Remark on linearity of the operator L, and linear combinations of solutions to homogeneous equation. • We have L[y1+y2]= L[ y1]+ L[y2], • for any constants and , and any twice differentiable functions y1 and y2 . • Theroem1. If y1 and y2 are solutions of the homogeneous equation • (HE): y+py+qy=0, then any linear combination y1+y2 of y1 and y2 is also a solution of (HE). 41 Consider an example • L[y] = y + 4y + 3y, We use the convention Dy = y , D2y = y , Dny = y(n) , • and rewrite L[y] = D2y + 4Dy +3y, or symbolically, L = D2 + 4D +3. Since formerly D2 + 4D +3 = (D + 3)(D + 1), we see that: • L[y] = (D + 3)(D + 1)[y]. The solutions for • L[y] = 0 are y1 = e-x , and y2 = e -3x. 42 Existence and Uniqueness of 2nd order equation • Theorem 2. Let p(x), q(x) and g(x) be continuous on an interval (a,b), and x0 (a,b). Then the I.V.P. y" p( x) y ' q( x) y g ( x) ; y ( x0 ) y0 , y ' ( x0 ) y1 , has a unique solution on the whole interval (a, b) for any choice of the initial values y0 and y1 . 43 Fundamental Solutions of Homogeneous Equations • Let us first define the notion of the Wronskian of two differentiable functions y1 and y2. The function W [ y1 , y2 ]( x) : y1 ( x) y2 ' ( x) y1 ' ( x) y2 ( x) is called the Wronskian of y1 and y2 . (in honor of H. Wronski, 1778 - 1853). Remark : this can be viewed as a 2x2 determinan t. 44 Fundamental solution set • A pair of solutions [y1, y2] of L[y] = 0, on (a,b) where L[y] = y+py+qy is called a fundamental solution set, if W[y1, y2](x0) 0 for some x0 (a,b) . A simple example: Consider • L[y] = y+9y. It is easily checked that y1 = cos 3x and y2 = sin 3x are solutions of L[y] = 0. Since the corresponding Wronskian W[y1, y2](x) = 3 0 , thus {cos 3x, sin 3x} forms a fundamental solution set to the homogenenous eq: y + 9y = 0. We see that any linear combination c1 y1 + c2 y2 also satisfies L[y] = 0. This is known as a general 45 solution Linear Independence, Fundamental set and Wronskian • Theorem. Let y1 and y2 be solutions to the equation y + py + qy = 0 on (a,b). Then the following statements are equivalent: • (A) {y1, y2} is a fundamental solution set on (a,b). • (B) y1 and y2 are linearly independent on (a,b). • (C) The W[y1, y2](x) is never zero on (a,b). • For the proof, we need some linear algebra, i.e. • Linearly dependent vectors,uniqueness theorem 46 etc... Reminder • First Hour Exam: • Date: June 15 (Friday) • Room: TBA 47 Homogeneous Linear Equations With Constant Coefficients • • • • • Recall: For equations of the form ay + by + cy = 0, by subsituting y = e r x, we obtain the auxiliary eq: ar2 + br + c = 0. If r1 and r2 are two distinct roots, then a general solution is of the form y = c1exp(r1x)+ c2exp(r2x), where c1 and c2 are arbitrary constants. 48 Repeated Roots • • • • • • • If in the above equation, r1 = r2 = r, then a general solution is of the form y = c1exp(rx) + c2x exp(rx), Example: consider the D.E. : y + 4y´ + 4 = 0. Its auxiliary equation is: r2 + 4r + 4 = 0, hence r = -2 is a double root, the general solution is y = c1e -2x + c2x e -2x, 49 Cauchy-Euler Equations • If an equation is of the form: ax2y + bxy´ + cy = h(x), a, b, c are constants, then by letting x = e t, we transform the original equation into:(with t as the independent variable), ay + (b-a)y´ + cy = h(e t). An equation with constant coefficients. Hence can be solved by the method of constant coefficients. The equation above is known as a Cauchy-Euler Equation. 50 Reduction of order • We know ,in general, a second order linear differential equation has two linearly independent solutions. If we already have one solution, how can we find the other one? • Let f(x) be a solution to y + p(x)y´ + q(x)y = 0. • We will try to find another solution of the form • y(x) = v(x)f(x), with v(x) a non-constant function. • Formerly, we have y ´ = v´f + vf ´, and y = … • set w = v ´, etc…, we obtain a separable eq. in w. 51 • Finally find v from w by integration. Example • • • • • • Given f(x) = x-1 is a solution to x2 y - 2xy´ -4y = 0, x > 0; find a second linearly independent solution. First write the D.E. in standard form. Next compute v. Finally, 2nd independent solution is y = v f. 52 Auxiliary Eq. With Complex Roots • If the auxiliary equation of a linear 2nd order D. E. with constant coefficents: ar2 + br + c = 0, has complex roots, (when b2 - 4ac < 0 ). i.e. • r1 = + i and r2 = - i, where and are real numbers, then the solutions are • y1 = e ( + i)x , and y2 = e ( - i)x. Since we know that e i x = cos x + i sin x , we simply take • y1 = e x cos x , and y2 = e x sin x as the two linearly independent solutions. 53 And the general solution is of the form • y(x) = c1 e x cos x + c2 e x sin x, where c1 and c2 are arbitrary constants. • Remark about complex solution: • z(x) = u(x) + iv(x) to L[z] = 0 and the fact that in this case, we also have L[u] = 0 and L[v] = 0. Thus the real part and the imaginary part of a complex solution to L[y] = 0 are also solutions of L[y] = 0. 54 Example • Consider the D.E. z"6 z '10 z 0. Clearly th e auxiliary equation is : r 2 6r 10 0. Since b 4ac (6) 4(1)(10) 4 0, we see 2 2 6 4 that the roots are r 3 i. 2 3x 3x Hence y1 e cos x, and y 2 e sin x. and the general solution is y c1e cos x c2 e sin x. 3x 3x 55 Nonhomogeneous Equation And the the method of Superposition • Let L be a linear operator of 2nd order, i.e. L= D2 + pD + q, and g 0. The equation: • L[y] = g, is called a Nonhomogeneous eq. • We wish to solve the equation L[y] = g , using a particular solution to L[y] = g , and a fundamental solution set to L[y] = 0. First let me introduce the concept of the method of superposition. 56 Theorem: Let y1 be a solution to the equation L[y] = g1, and let y2 be a solution to the equation L[y] = g2, where g1 and g2 are two functions. Then for any two constants c1 and c2, the linear combination c1 y1 + c2 y2 is a solution to the equation L[y] = c1 g1 + c2 g2 . (This is known as the Superposition principle). 57 Proof 58 Representation Theorem of L[y] = g. • Theorem: Let yp(x) be a particular solution to the nonhomogeneous equation (*) L[y] = g(x), where L[y]= y + p(x) y + q(x) y , on the interval (a,b) and let y1(x) and y2(x) be a fundamental solution set of L[y] = 0 on the interval (a,b). Then every solution of (*) can be written in the form • (**) y(x) = yp(x) + c1 y1(x) +c2 y2(x) . This is known as the general solution to (*). 59 Example • Given that yp(x) = x2 is a particular solution of the equation: • (*) y - y = 2 - x2, • find a general solution of (*). • Note the auxiliary equation is r 2 - 1 = 0. It follows that a general solution of (*) is of the form y = x2 + c1e-x + c2e x. 60 Superposition Principle & the Method of Undetermined coefficients. • Example: Find a general solution to the D.E. y"3y x e . 2 x • Step 1: We first consider the associated homogenous equation: y"3 y 0 61 Step 2: Find particular solution to the Non-homogenous equation using the Superposition Principle • There are “2” equations: y"3 y x 2 y"3 y e . x 62 To find a particular solution to each of the above equations • We use the method of undetermined coefficients, that is: for the first equation, we try yp = ax2 + bx + c, and • For the second equation, we try yp = Aex. • If any term in the trial expression for yp is a solution to the corresponding homogeneous equation, then we replace yp by x yp, etc…. See table 4.1 on Page 208 of your book. 63 Next we present a more general method, known as: • The method of variation of parameters, 64 The Method of Variation of Parameters • Consider the non-homogenous linear second order differential equation : y" p( x ) y ' q ( x ) y g ( x ). ..........(1) Let { y1 ( x ), y2 ( x )} be a fundamental solution set for the corresponding homogeneous equation y" p( x ) y ' q ( x ) y 0. We try particular solution of the equation (1) in the form: y p v1 ( x ) y1 ( x ) v2 ( x ) y2 ( x ). 65 Where v1 and v2 are functions to be determined. We obtain: • two equations (by avoiding 2nd order derivatives for the unknows and from L[yp]=g): v1 ' y1 v2 ' y2 0, and v1 ' y1 ' v2 ' y2 ' g. solving these two equations simultaneuously, we get g ( x ) y2 ( x ) g ( x ) y1 ( x ) v1 '( x ) , v2 '( x ) . W[ y1 , y2 ]( x ) W[ y1 , y2 ]( x ) 66 Where v1 and v2 are functions to be determined. We obtain: • two equations (by avoiding 2nd order derivatives for the unknows and from L[yp]=g): v1 ' y1 v2 ' y2 0, and v1 ' y1 ' v2 ' y2 ' g. solving these two equations simultaneuously, we get g ( x ) y2 ( x ) g ( x ) y1 ( x ) v1 '( x ) , v2 '( x ) . W[ y1 , y2 ]( x ) W[ y1 , y2 ]( x ) 67 Finally, solution is found by integration. • Example: Find a general solution t o the D.E. y"4 y '4 y e 2 x ln x . 68 69