Suppose that the three numbers rl , r2 , and r 3 are distinct. Show that the three

Elementary Differential Equations | 6th Edition | ISBN: 9780132397308 | Authors: C. Henry Edwards David E. Penney

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" - y = 0; Yl = eX, Yz = e-x; yeO) = 0, y' (0) = 5

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" - 9y = 0; Y l = e3x, Yz = e-3x; yeO) = -1, y'(0) = 1 5

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + 4y = 0; Yl = cos 2x , Yz = sin 2x ; yeO) = 3, y'(O) = 8

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + 25y = 0; Yl = cos 5x, yz = sin 5x ; yeO) = 10, y'(0) = -10

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" - 3y' +2y = 0; Yl = eX, Yz = e Z x; yeO) = 1, y'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + y' - 6y = 0; Y l = e Z X, Yz = e-3x; yeO) = 7, y'(0) = -1

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + y' = 0; YI = I, Y2 = e-x ; yeO) = -2, y'(O) = 8

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" - 3y' = 0; YI = I, Y2 = e3x; yeO) = 4, y'(O) = -2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + 2y' + y = 0; YI = e-x, Y2 = xe-x ; yeO) = 2, y'(O) = - I

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" - lOy' + 25y = 0; YI = e5x, Y2 = xe5x; yeO) = 3, y'(O) = 13

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" -2y' +2y = 0; YI = eX cos x, Y2 = eX sin x; yeO) = 0, y'(O) =

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + 6y' + By = 0; YI = e-3x cos2x, Y2 = e-3x sin2x; yeO) = 2, y'(O) =

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. x2y" - 2xy' + 2y = 0; YI = X , Y2 = x2; y(1) = 3, y'(1) = I

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. x2y" + 2xy' - 6y = 0; YI = x2, Y2 = x-3 ; y(2) = 10, y'(2) = 15

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. x2y" - xy' + y = 0; YI = X , Y2 = x ln x; y(1) = 7, y'(I) =

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 16, a homogeneous second-order linear differential equation, two functions Yl and yz, and a pair of initial conditions are given. First verify that Yl and Yz are solutions of the differential equation. Then find a particular solution of the form Y = Cl Yl + czYz that satisfies the given initial conditions. Primes denote derivatives with respect to x. x2y" + xy' + y = 0; YI = cos(1n x), Y2 = sin(1n x); y(1) = 2, y'(1) = 3

Chapter 2: Problem 2 Elementary Differential Equations 6

Show that y = I /x is a solution of y' + y2 = 0, but that if c =1= 0 and c =1= I, then y = c / x is not a solution.

Chapter 2: Problem 2 Elementary Differential Equations 6

Show that y = x3 is a solution of yy" = 6x4, but that if c 2 =1= 1, then y = cx3 is not a solution.

Chapter 2: Problem 2 Elementary Differential Equations 6

Show that YI == I and Y2 = ..Ji are solutions of yy" + (y')2 = 0, but that their sum y = YI + Y2 is not a solution.

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = rr, g(x) = cos2 X + sin2 x

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = x3, g(x) = x2 1x l

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = I + x, g(x) = I + Ix l

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = xex, g(x) = Ix lex

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = sin2 x, g(x) = 1 - cos2x

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = eX sin x, g(x) = eX cos x

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine whether the pairs of functions in Problems 20 through 26 are linearly independent or linearly dependent on the real line. f(x) = 2 cos x + 3 sin x, g(x) = 3 cos x - 2 sin x

Chapter 2: Problem 2 Elementary Differential Equations 6

Let yp be a particular solution of the nonhomogeneous equation y" + py' + qy = f(x) and let Yc be a solution of its associated homogeneous equation. Show that y = Yc + YP is a solution of the given nonhomogeneous equation.

Chapter 2: Problem 2 Elementary Differential Equations 6

With YP = I and Yc = CI COS X + C2 sin x in the notation of Problem 27, find a solution of y" + y = I satisfying the initial conditions yeO) = -1 = y'(O).

Chapter 2: Problem 2 Elementary Differential Equations 6

Show that YI = x2 and Y2 = x3 are two different solutions of x2y" - 4xy' + 6y = 0, both satisfying the initial conditions yeO) = 0 = y'(O). Explain why these facts do not contradict Theorem 2 (with respect to the guaranteed uniqueness).

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) Show that YI = x3 and Y2 = Ix3 1 are linearly independent solutions on the real line of the equation x2 y" -3x y' + 3 y = O. (b) Verify that W (YI , Y2 ) is identically zero. Why do these facts not contradict Theorem 3?

Chapter 2: Problem 2 Elementary Differential Equations 6

Show that YI = sin x2 and Y2 = cos x2 are linearly independent functions, but that their Wronskian vanishes at x = O. Why does this imply that there is no differential equation of the form y" + p(x)y' + q(x)y = 0, with both p and q continuous everywhere, having both YI and Y2 as solutions?

Chapter 2: Problem 2 Elementary Differential Equations 6

Let YI and Y2 be two solutions of A(x)y" + B(x)y' + C(x)y = 0 on an open interval 1 where A, B, and C are continuous and A(x) is never zero. (a) Let W = W(YI , Y2). Show that Then substitute for Ay and Ay;' from the original differential equation to show that dW A(x)- = -B(x)W(x). dx (b) Solve this first-order equation to deduce Abel's formula W (x) = K exp (-f B(x) dX) , A(x) where K is a constant. (c) Why does Abel's formula imply that the Wronskian W (YI , Y2 ) is either zero everywhere or nonzero everywhere (as stated in Theorem 3)?

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. y" - 3y' + 2y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. y" + 2y' - 15y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. y" +5y' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 2y" + 3y' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 2y" - y' - y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 4y" + 8y' + 3y =

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 4y" + 4y' + y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 9y" - 12y' +4y =

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 6y" - 7y' - 20y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply Theorems 5 and 6 to find general solutions of the differential equations given in Problems 33 through 42. Primes denote derivatives with respect to x. 35y" - y' - 12y =

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay" + by' + cy = 0 with constant coefficients. Find such an equation. y(x) = CI + C2e- 10

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay" + by' + cy = 0 with constant coefficients. Find such an equation.

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay" + by' + cy = 0 with constant coefficients. Find such an equation. y(x) = cle-lO x + c 2xe- I Ox

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay" + by' + cy = 0 with constant coefficients. Find such an equation. y(x) = cle lO x + C2el OOx

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay" + by' + cy = 0 with constant coefficients. Find such an equation. y(x) = CI + C2X

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 43 through 48 gives a general solution y(x) of a homogeneous second-order differential equation ay" + by' + cy = 0 with constant coefficients. Find such an equation. y(x) = eX (c1exJ2 + c2 e-xJ2)

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 49 and 50 deal with the solution curves of y" + 3 y' + 2y = 0 shown in Figs. 2. 1.6 and 2. 1. 7. Find the highest point on the solution curve with yeO) = I and y' (0) = 6 in Fig. 2. 1 .6.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 49 and 50 deal with the solution curves of y" + 3 y' + 2y = 0 shown in Figs. 2. 1.6 and 2. 1. 7. Find the third-quadrant point of intersection of the solution curves shown in Fig. 2. 1.7.

Chapter 2: Problem 2 Elementary Differential Equations 6

A second-order Euler equation is one of the fonn conclude that a general solution of the Euler equation in (22) is y(x) = C1Xrl + C2X r2 . ax2y" + bxy' + cy = 0 (22) where a, b, c are constants. (a) Show that if x > 0, then the substitution v = ln x transfonns Eq. (22) into the constant-coefficient linear equation with independent variable v. (b) If the roots rl and r2 of the characteristic equation of Eq. (23) are real and distinct, conclude that a general solution of the Euler equation in (22) is y(x) = C1Xrl + C2X r2.

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52- 56. x2y" + xy' - y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52- 56. x2y" + 2xy' - 12y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52- 56. 4x2y" + 8xy' - 3y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52- 56. x2y" + xy' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52- 56. x2y" -3xy' + 4y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = 2x, g(x) = 3x2, h ex) = 5x - 8x2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = 5, g(x) = 2 - 3x2, h ex) = 10 + 15x2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = 0, g(x) = sin x, h ex) = eX

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = 17, g(x) = 2 sin2 x, h ex) = 3 cos2 x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = 17, g(x) = cos2 x, h ex) = cos 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 1 through 6, show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = eX, g(x) = cosh x, hex) = sinh x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 7 through 12, use the Wronskian to prove that the given functions are linearly independent on the indicated interval. f(x) = 1, g(x) = x, h ex) = x2 ; !he real line

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 7 through 12, use the Wronskian to prove that the given functions are linearly independent on the indicated interval. f(x) = eX, g(x) = e2x, h ex) = e3x; !he real line

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 7 through 12, use the Wronskian to prove that the given functions are linearly independent on the indicated interval. f(x) = eX, g(x) = cos x, hex) = sin x ; the real line

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 7 through 12, use the Wronskian to prove that the given functions are linearly independent on the indicated interval. f(x) = eX, g(x) = x-2, hex) = x-2 ln x ; x > 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 7 through 12, use the Wronskian to prove that the given functions are linearly independent on the indicated interval. f(x) = x, g(x) = xeX, h ex) = x2ex ; the real line

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 7 through 12, use the Wronskian to prove that the given functions are linearly independent on the indicated interval. f(x) = x, g(x) = cos(ln x), h ex) = sin (In x); x > 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y(3) +2y" -y' -2y = 0; yeO) = 1, leO) = 2, y " (O) = 0; YI = eX, Y2 = e-X, Y3 = e-2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y(3) - 6y" + l ly' - 6y = 0; yeO) = 0, y' (O) = 0, y"(O) = 3; YI = eX, Y2 = e 2x, Y3 = e3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y< 3) - 3y" + 3y' - y = 0; yeO) = 2, y'(O) = 0, y"(O) = 0; YI = eX, Y2 = xex, Y3 = x2ex

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y( 3) _5y" +8y' -4y = 0; yeO) = 1, y'(O) = 4, y"(O) = 0; YI = eX, Y2 = e 2x, Y3 = xe2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y(3) + 9y' = 0; yeO) = 3, y'(O) = -1, y"(O) = 2; YI = 1, Y2 = cos 3x, Y3 = sin 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. y(3) - 3y" +4y' - 2y = 0; yeO) = 1, y'(O) = 0, y"(O) = 0; YI = eX, Y2 = eX cos x, Y3 = eX sin x.

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. x3y( 3) - 3x2y" + 6xy' - 6y = 0; y(l) = 6, y'(1) = 14, y"(1) = 22; YI = X, Y2 = x2, Y3 = x3

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. x3y(3) + 6x2y" + 4xy' - 4y = 0; y(1) = 1 , y' (1) = 5, y"(I) = -11 ; YI = X, Y2 = X-2, Y3 = x-2 In x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 24, a nonhomogeneous differential equation, a complementary solution Yo and a particular solution yp are given. Find a solution satisfying the given initial conditions. y" + y = 3x ; yeO) = 2, y'(O) = -2; Yc = CI cos x + C2 sin x ; YP = 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 24, a nonhomogeneous differential equation, a complementary solution Yo and a particular solution yp are given. Find a solution satisfying the given initial conditions. y" - 4y = 12; yeO) = 0, y'(O) = 10; Yc = cle 2x + C2 e-2 x; YP = -3

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 24, a nonhomogeneous differential equation, a complementary solution Yo and a particular solution yp are given. Find a solution satisfying the given initial conditions. y" - 2y' - 3y = 6; yeO) = 3, y'(O) = 1 1 ; Yc = cle-x + C2 e3x; YP = -2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 24, a nonhomogeneous differential equation, a complementary solution Yo and a particular solution yp are given. Find a solution satisfying the given initial conditions. y" - 2y' + 2y = 2x ; yeO) = 4, y'(O) = 8; Yc = clex cos x + C2 ex sin x ; YP = x + 1

Chapter 2: Problem 2 Elementary Differential Equations 6

Let Ly = y" + py' + qy. Suppose that YI and Y2 are two functions such that LYI = f(x) and LY2 = g (x). Show that their sum y = YI + Y2 satisfies the nonhomogeneous equation Ly = f(x) + g(x).

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) Find by inspection particular solutions of the two nonhomogeneous equations y" + 2y = 4 and y" + 2y = 6x. (b) Use the method of Problem 25 to find a particular solution of the differential equation y" + 2y = 6x + 4.

Chapter 2: Problem 2 Elementary Differential Equations 6

Prove directly that the functions fl (x) == 1 , hex) = x, and hex) = x2 are linearly independent on the whole real line. (Suggestion: Assume that CI + C2 X + C3 x2 = O. Differentiate this equation twice, and conclude from the equations you get that CI = C2 = C3 = 0.)

Chapter 2: Problem 2 Elementary Differential Equations 6

Generalize the method of Problem 27 to prove directly that the functions are linearly independent on the real line.

Chapter 2: Problem 2 Elementary Differential Equations 6

Use the result of Problem 28 and the definition of linear independence to prove directly that, for any constant r, the functions fo(x) = e'x, !l ex) = xe'x' are linearly independent on the whole real line.

Chapter 2: Problem 2 Elementary Differential Equations 6

Verify that YI = x and Y2 = x2 are linearly independent solutions on the entire real line of the equation x2y" - 2xy' + 2y = 0, but that W (x , x2) vanishes at x = O. Why do these observations not contradict part (b) of Theorem 3?

Chapter 2: Problem 2 Elementary Differential Equations 6

This problem indicates why we can impose only n initial conditions on a solution of an nth-order linear differential equation. (a) Given the equation y" + py' + qy = 0, explain why the value of y"(a) is determined by the values of yea) and y'(a). (b) Prove that the equation y" - 2y' - 5y = 0 has a solution satisfying the conditions yeO) = 1, y'(O) = 0, and y"(O) = c if and only if C = 5.

Chapter 2: Problem 2 Elementary Differential Equations 6

Prove that an nth-order homogeneous linear differential equation satisfying the hypotheses of Theorem 2 has n linearly independent solutions YI , Y2 , ... , Yn. (Suggestion: Let Yj be the unique solution such that ( iI ) ( ) - Yj a - 1 and Yj ( k) (a) = 0 ifk i - I .)

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that the three numbers rl , r2 , and r 3 are distinct. Show that the three functions exp(rlx), exp(r2 x), and exp(r 3 x) are linearly independent by showing that their Wronskian is nonzero for all x

Chapter 2: Problem 2 Elementary Differential Equations 6

Assume as known that the Vandermonde determinant rl r2 rn v= r 2 I r 2 2 r 2 n nI rl nI r 2 r:- I is nonzero if the numbers rl, r 2 , ... , rn are distinct. Prove by the method of Problem 33 that the functions fi (x) = exp(rjx), 1 i n are linearly independent.

Chapter 2: Problem 2 Elementary Differential Equations 6

According to Problem 32 of Section 2. 1, the Wronskian W (YI , Y2) of two solutions of the second-order equation is given by Abel's's formula W(x) = K exp (- f PI (X) dX) for some constant K. It can be shown that the Wronskian of n solutions YI> Y2, ... , Yn of the nth-order equation y( n ) + PI (x)y.

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that one solution YI (x) of the homogeneous second-order linear differential equation Y" + p(x)y' + q (x)y = 0 (18) is known (on an interval I where P and q are continuous functions). The method of reduction of order consists of substituting Y2 (X) = V (X)YI (x) in (18) and attempting to determine the function vex) so that Y2(X) is a second linearly independent solution of (18). After substituting y = V (X)YI (x) in Eq. (18), use the fact that Yl (x) is a solution to deduce that (19) If YI (x) is known, then (19) is a separable equation that is readily solved for the derivative v' (x) of vex). Integration of v'(x) then gives the desired (nonconstant) function vex)

Chapter 2: Problem 2 Elementary Differential Equations 6

Before applying Eq. (19) with a given homogeneous second-order linear differential equation and a known solution Yl (x), the equation must first be written in the form of (18) with leading coefficient I in order to correctly determine the coefficient function p(x). Frequently it is more convenient to simply substitute Y = V (X) Yl (X) in the given differential equation and then proceed directly to find vex). Thus, starting with the readily verified solution YI (x) = x3 of the equation substitute Y = vx3 and deduce that xv" + v' = O. Thence solve for vex) = C In x, and thereby obtain (with C = 1) the second solution Y2(X) = x3 ln x.

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 38 through 42, a differential equation and one solution Yl are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution Y2 x 2 y" + xy' - 9y = 0 (x > 0) ; YI (x) = x3

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 38 through 42, a differential equation and one solution Yl are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution Y2 4y" - 4y' + Y = 0; Yl (x) = ex/2

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 38 through 42, a differential equation and one solution Yl are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution Y2 x 2 y" - x(x + 2)y' + (x + 2)y = 0 (x > 0); YI (x) = x

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 38 through 42, a differential equation and one solution Yl are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution Y2 (x + l )y" - (x + 2)y' + y=0 (x > - 1 ); Yl (x) = ex

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 38 through 42, a differential equation and one solution Yl are given. Use the method of reduction of order as in Problem 37 to find a second linearly independent solution Y2 ( 1 - x 2)y" + 2xy' - 2y = 0 (-1 < x < 1); Yl (X) = x

Chapter 2: Problem 2 Elementary Differential Equations 6

First note that YI (x) = x is one solution of Legendre's equation of order 1, (1 - x2 )y" - 2xy' + 2y = O. Then use the method of reduction of order to derive the second solution x 1 +x Y2(X) = 1 - - In-- (for -l < x < 1). 2 I - x

Chapter 2: Problem 2 Elementary Differential Equations 6

First verify by substitution that Yl (x) = x - 1/ 2 cos x is one solution (for x > 0) of Bessel's equation of order , Then derive by reduction of order the second solution Y2(X) = X- 1/ 2 sin x.

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y" -4y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. 2y" - 3y' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y" + 3y' - lOy = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. 2y" - 7y' + 3y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y" + 6y' + 9y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y" + 5 y' + 5 y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. 4y" - 12y' +9y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y" - 6y' + 13y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y" + 8y' + 25y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. 5y(4) + 3y(3) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) -8y(3) + 16y" = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) -3y(3) + 3y" - y' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. 9y(3) + 12y" + 4y' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) + 3y" -4y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) - 8y" + 16y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) + 18y" + 81y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. 6y(4) + lly" + 4y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) = 16y

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(3) + y" - y' - y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the general solutions of the differential equations in Problems 1 through 20. y(4) + 2y(3) + 3y" + 2y' + y = 0 (Suggestion: Expand (r2 + r + 1)2.)

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems given in Problems 21 through 26. y" -4y' + 3y = 0; yeO) = 7, y'(O) = 11

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems given in Problems 21 through 26. 9y" + 6y' + 4y = 0; yeO) = 3, y'(O) = 4

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems given in Problems 21 through 26. y" - 6y' + 25y = 0; yeO) = 3, y'(O) = 1

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems given in Problems 21 through 26. 2y(3) -3y" - 2y' = 0; yeO) = 1, y'(O) = -1, y"(O) = 3

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems given in Problems 21 through 26. 3y(3) + 2y" = 0; yeO) = -1, y'(O) = 0, y"(O) = 1

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems given in Problems 21 through 26. y(3) + lOy" + 25y' = 0; yeO) = 3, y'(O) = 4, y"(O) = 5

Chapter 2: Problem 2 Elementary Differential Equations 6

Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division.. y(3) + 3y" -4y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division. 2y(3) - y" -5y' - 2y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division. y(3) + 27y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division. y(4) -y(3) + y" - 3y' - 6y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division. y(3) + 3y" + 4y' - 8y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find general solutions of the equations in Problems 27 through 32. First find a small integral root of the characteristic equation by inspection; then factor by division. y(4) + y(3) - 3y" -5y' - 2y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 33 through 36, one solution of the differential equation is given. Find the general solution. y(3) + 3y" -54y = 0; Y = e3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 33 through 36, one solution of the differential equation is given. Find the general solution. 3y(3) -2y" + 12y' - 8y = 0; y = e 2 xf3

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 33 through 36, one solution of the differential equation is given. Find the general solution. 6y(4) + 5y(3) + 25y" + 20y' + 4y = 0; Y = cos 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 33 through 36, one solution of the differential equation is given. Find the general solution. 9y(3) + lly" + 4y' - 14y = 0; y = e-x sin x

Chapter 2: Problem 2 Elementary Differential Equations 6

Find a function y(x) such that y(4) (X) = y(3) (X) for all x and yeO) = 18, y'(O) = 12, y"(O) = 13, and y(3)(0) = 7.

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problem y( 3) _ 5y" + 100y' -500y = 0; yeO) = 0, y' (0) = 10, y" (0) = 250 given that Yl (x) = e5x is one particular solution of the differential equation.

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution. y(x) = (A + Bx+ Cx2)e2 x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution. y(x) = Ae2 x + B cos 2x + C sin 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution. y(x) = A cos 2x + B sin 2x + C cosh 2x + D sinh 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 39 through 42, find a linear homogeneous constant-coefficient equation with the given general solution. y(x (A+Bx+Cx2) cos 2x+(D+Ex+Fx2) sin 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 43 through 47 pertain to the solution of differential equations with complex coefficients. (a) Use Euler's formula to show that every complex number can be written in the form rei 8, where r 0 and -n < e n . (b) Express the numbers 4, -2, 3i, 1 + i, and -1 + i../3 in the form rei 8 (c) The two square roots of rei 8 are ..jre i 8/2 Find the square roots of the numbers 2 - 2i../3 and -2 + 2i../3.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 43 through 47 pertain to the solution of differential equations with complex coefficients. Use the quadratic formula to solve the following equations. Note in each case that the roots are not complex conjugates. (a) x 2 + ix + 2 = 0 (b) x 2 - 2ix + 3 = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 43 through 47 pertain to the solution of differential equations with complex coefficients. Find a general solution of y" -2iy' + 3y = O.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 43 through 47 pertain to the solution of differential equations with complex coefficients. Find a general solution of y" - iy' + 6y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 43 through 47 pertain to the solution of differential equations with complex coefficients. Find a general solution of y" = (-2 + 2i../3) y.

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problem y( 3) = y; yeO) = 1, y'(O) = y"(O) = O. (Suggestion: Impose the given initial conditions on the general solution y(x) = Aex + Be"x + CefJx, where ex and fJ are the complex conjugate roots of r3 -1 = 0, to discover that1 ( x.J3) y(x) = "3 eX + 2e-x/ 2 cos 2 is a solution.)

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problem /4) = y( 3) + y" + y' + 2y; yeO) = /(0) = y"(O) = 0, 2y( 3) (0) = 30.

Chapter 2: Problem 2 Elementary Differential Equations 6

The differential equation y" + (sgn x)y = 0 has the discontinuous coefficient function { + 1 if x> 0, sgn x = -1 if x < O. (25) Show that Eq. (25) nevertheless has two linearly independent solutions Yl (x) and Y2 (X) defined for all x such that Each satisfies Eq. (25) at each point x #- 0; Each has a continuous derivative at x = 0; Yl (0) = y(O) = 1 and Y2 (0) = y; (0) = O. (Suggestion: Each Yi (x) will be defined by one formula for x < 0 and by another for x 0.) The graphs of these two solutions are shown in Fig. 2.3.2.

Chapter 2: Problem 2 Elementary Differential Equations 6

According to Problem 51 in Section 2. 1, the substitution v = In x (x > 0) transforms the second-order Euler equation ax2y" + bxy' + cy = 0 to a constant-coefficient homogeneous linear equation. Show similarly that this same substitution transforms the third-order Euler equation ax3ylll + bx2y" + cxy' + dy = 0 (where a, b, c, d are constants) into the constantcoefficient equation y y a- + (b - 3a)- + (c - b+2a)- +dy =0. dv3 dv2 dv

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. x2y" + xy' + 9y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. x2y" + 7xy' + 25y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. X3ylll + 6x2y" + 4xy' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. X3ylll -x2y" + xy' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. x3yl/l + 3x2y" + xy' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. x3y'" - 3x2y" + xy' = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Make the substitution v = In x of Problem 51 to find general solutions (for x > 0) of the Euler equations in Problems 52 through 58. X3ylll + 6x2y" + 7xy' + y = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine the period and frequency of the simple harmonic motion of a 4-kg mass on the end of a spring with spring constant 16 N/m.

Chapter 2: Problem 2 Elementary Differential Equations 6

Determine the period and frequency of the simple harmonic motion of a body of mass 0.75 kg on the end of a spring with spring constant 48 N/m.

Chapter 2: Problem 2 Elementary Differential Equations 6

A mass of 3 kg is attached to the end of a spring that is stretched 20 cm by a force of 15 N. It is set in motion with initial position Xa = 0 and initial velocity Va = - 10 m/s. Find the amplitude, period, and frequency of the resulting motion.

Chapter 2: Problem 2 Elementary Differential Equations 6

A body with mass 250 g is attached to the end of a spring that is stretched 25 cm by a force of 9 N. At time t = 0 the body is pulled 1 m to the right, stretching the spring, and set in motion with an initial velocity of 5 m/s to the left. (a) Find x(t) in the form C cos(wot - a). (b) Find the amplitude and period of motion of the body.

Chapter 2: Problem 2 Elementary Differential Equations 6

Two pendulums are of lengths LJ and L2 and-when located at the respective distances RI and R2 from the center of the earth-have periods PI and P2 . Show that PI RI....;r; P2 = R2 .

Chapter 2: Problem 2 Elementary Differential Equations 6

A certain pendulum keeps perfect time in Paris, where the radius of the earth is R = 3956 (mi). But this clock loses 2 min 40 s per day at a location on the equator. Use the result of Problem 5 to find the amount of the equatorial bulge of the earth.

Chapter 2: Problem 2 Elementary Differential Equations 6

A pendulum of length 100. 10 in., located at a point at sea level where the radius of the earth is R = 3960 (mi), has the same period as does a pendulum of length 100.00 in. atop a nearby mountain. Use the result of Problem 5 to find the height of the mountain.

Chapter 2: Problem 2 Elementary Differential Equations 6

Most grandfather clocks have pendulums with adjustable lengths. One such clock loses 10 min per day when the length of its pendulum is 30 in. With what length pendulum will this clock keep perfect time?

Chapter 2: Problem 2 Elementary Differential Equations 6

Derive Eq. (5) describing the motion of a mass attached to the bottom of a vertically suspended spring. (Suggestion: First denote by x(t) the displacement of the mass below the unstretched position of the spring; set up the differential equation for x. Then substitute y = x - Xo in this differential equation.)

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider a floating cylindrical buoy with radius r, height h, and uniform density p 0.5 (recall that the density of water is 1 g/cm3 ). The buoy is initially suspended at rest with its bottom at the top surface of the water and is released at time t = O. Thereafter it is acted on by two forces: a downward gravitational force equal to its weight mg = nr2hg and (by Archimedes' principle of buoyancy) an upward force equal to the weight nr2xg of water displaced, where x = x(t) is the depth of the bottom of the buoy beneath the surface at time t (Fig. 2.4. 1 2). Conclude that the buoy undergoes simple harmonic motion around its equilibrium position Xe = ph with period p = 2n.J ph/g. Compute p and the amplitude of the motion if p = 0.5 g/cm3 , h = 200 cm, and g = 980 cm/s2 .

Chapter 2: Problem 2 Elementary Differential Equations 6

A cylindrical buoy weighing 100 lb (thus of mass m 3. 1 25 slugs in ft-lb-s (fps) units) floats in water with its axis vertical (as in Problem 10). When depressed slightly and released, it oscillates up and down four times every 10 s. Assume that friction is negligible. Find the radius of the buoy.

Chapter 2: Problem 2 Elementary Differential Equations 6

Assume that the earth is a solid sphere of uniform density, with mass M and radius R = 3960 (mi). For a particle of mass m within the earth at distance r from the center of the earth, the gravitational force attracting m toward the center is Fr = -GMr m/r2, where Mr is the mass of the part of the earth within a sphere of radius r. (a) Show that Fr = -GMmr/R3 (b) Now suppose that a small hole is drilled straight through the center of the earth, thus connecting two antipodal points on its surface. Let a particle of mass m be dropped at time t = 0 into this hole with initial speed zero, and let r(t) be its distance from the center of the earth at time t (Fig. 2.4. 1 3). Conclude from Newton's second law and part (a) that r"(t) = -er(t), where k 2 = GM/R3 = giRo (c) Take g = 32.2 ft/S2, and conclude from part (b) that the particle undergoes simple harmonic motion back and forth between the ends of the hole, with a period of about 84 min. (d) Look up (or derive) the period of a satellite that just skims the surface of the earth; compare with the result in part (c). How do you explain the coincidence? Or is it a coincidence? (e) With what speed (in miles per hour) does the particle pass through the center of the earth? (1) Look up (or derive) the orbital velocity of a satellite that just skims the surface of the earth; compare with the result in part (e). How do you explain the coincidence? Or is it a coincidence?

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that the mass in a mass-spring-dashpot system with m = 10, c = 9, and k = 2 is set in motion with x(O) = 0 and x'(O) = 5. (a) Find the position function x(t) and show that its graph looks as indicated in Fig. 2.4. 14. (b) Find how far the mass moves to the right before starting back toward the origin.

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that the mass in a mass-spring-dashpot system with m = 25, c = 10, and k = 226 is set in motion with x(O) = 20 and x'(O) = 41. (a) Find the position function x(t) and show that its graph looks as indicated in Fig. 2.4. 15. (b) Find the pseudoperiod of the oscillations and the equations of the "envelope curves" that are dashed in the figure.

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = !, c = 3, k = 4; Xo = 2, Vo = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = 3, c = 30, k = 63; Xo = 2, Vo = 2

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = I, c = 8, k = 16; Xo = 5, Vo = -10

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = 2, c = 12, k = 50; Xo = 0, Vo = -8

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = 4, c = 20, k = 169; Xo = 4, Vo = 16

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = 2, c = 16, k = 40; Xo = 5, Vo = 4

Chapter 2: Problem 2 Elementary Differential Equations 6

The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position Xo and initial velocity Vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or under damped. If it is underdamped, write the position function in the form x(t) = C1 e-pt cOS(Wl t-al ). Also, find the undamped position function u(t) = Co cos(wot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs ofx(t) and u(t). m = I, c = 10, k = 125; Xo = 6, Vo = 50

Chapter 2: Problem 2 Elementary Differential Equations 6

A 1 2-lb weight (mass m = 0.375 slugs in fps units) is attached both to a vertically suspended spring that it stretches 6 in. and to a dashpot that provides 3 Ib of resistance for every foot per second of velocity. (a) If the weight is pulled down I ft below its static equilibrium position and then released from rest at time t = 0, find its position function x(t). (b) Find the frequency, time-varying amplitude, and phase angle of the motion.

Chapter 2: Problem 2 Elementary Differential Equations 6

This problem deals with a highly simplified model of a car of weight 3200 Ib (mass m = 100 slugs in fps units). Assume that the suspension system acts like a single spring and its shock absorbers like a single dashpot, so that its vertical vibrations satisfy Eq. (4) with appropriate values of the coefficients. (a) Find the stiffness coefficient k of the spring if the car undergoes free vibrations at 80 cycles per minute (cycles/min) when its shock absorbers are disconnected. (b) With the shock absorbers connected, the car is set into vibration by driving it over a bump, and the resulting damped vibrations have a frequency of 78 cycles/min. After how long will the time-varying amplitude be I % of its initial value?

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Critically damped) Show in this case that x(t) = (xo + vot + pxot)e-pt .

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Critically damped) Deduce from Problem 24 that the mass passes through x = 0 at some instant t > 0 if and only if Xo and Vo + pXo have opposite signs.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Critically damped) Deduce from Problem 24 that x (t) has a local maximum or minimum at some instant t > 0 if and only if Vo and Vo + pXo have the same sign.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Overdamped) Show in this case that where r1, r2 = -p J p2 - wB and y = (rl -r2)/2 > O.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Overdamped) If Xo = 0, deduce from Problem 27 that x(t) = Vo ept sinh y t

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Overdamped) Prove that in this case the mass can pass through its equilibrium position x = 0 at most once.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Underdamped) Show that in this case_ ( Vo + pXo . ) x(t) = e pt xO COS Wl t +WI sm wl t

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Underdamped) If the damping constant c is small in comparison with .,/ 8mk, apply the binomial series to show that WI WO( I - ) . 8mk

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Underdamped) Show that the local maxima and minima of occur where tan(wl t - a) = _.!!..... WI Conclude that t 2 -t l = 2n/wl if two consecutive maxima occur at times tl and t 2 .

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Underdamped) Let XI and X2 be two consecutive local maximum values of X (t). Deduce from the result of Problem 32 that In XI = 2np X2 WIThe constant = 2n P/WI is called the logarithmic decrement of the oscillation. Note also that c = mWI /n because p = c/(2m).

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 24 through 34 deal with a mass-spring-dashpot system having position function x(t) satisfying Eq. (4). We write Xo = x(O) and Vo = x'(O) and recall that p = c/(2m), w5 = kim, and wr = w5 - p2. The system is critically damped, overdamped, or under damped, as specified in each problem. (Underdamped) A body weighing 100 Ib (mass m =3. 1 25 slugs in fps units) is oscillating attached to a spring and a dashpot. Its first two maximum displacements of 6.73 in. and 1 .46 in. are observed to occur at times 0.34 s and 1.17 s, respectively. Compute the damping constant (in pound-seconds per foot) and spring constant (in pounds per foot).

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that m = 1, c = 2, and k = 1 in Eq. (26). Show that the solution with x(O) = 0 and x'(O) = 1 is XI (t) = te-t .

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that m = 1 and c = 2 but k = 1 - 1O2n . Show that the solution of Eq. (26) with x(O) = 0 and x'(O) = 1 is X2 (t) = lO ne-t sinh lOn t.

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that m = 1 and c = 2 but that k = 1 + 1O2n . Show that the solution of Eq. (26) with x(O) = 0 and x'(O) = 1 is

Chapter 2: Problem 2 Elementary Differential Equations 6

Whereas the graphs of x, (t) and X2 (t) resemble those shown in Figs. 2.4.7 and 2.4.8, the graph of X3 (t) exhibits damped oscillations like those illustrated in Fig. 2.4.9, but with a very long pseudoperiod. Nevertheless, show that for each fixed t > 0 it is true that lim X2 (t) = lim X3 (t) = x,(t). noo noo Conclude that on a given finite time interval the three solutions are in "practical" agreement if n is sufficiently large.

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + 1 6y = e3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" - y' - 2y=3x + 4

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" - y' - 6y = 2 sin 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. 4y" + 4y' + y = 3xex

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + y' + y = sin2 x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. 2y" + 4y' + 7y = x2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" - 4y = sinh x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" - 4y = cosh 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + 2y' - 3y = I + xeX

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + 9y = 2 cos 3x + 3 sin 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y(3) + 4y' = 3x - I

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y(3) + y' = 2 - sin x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + 2y' + 5y = eX sin x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y(4) - 2y" + y = xeX

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y (5) + 5y(4) - y = 17

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + 9y = 2x2e3x + 5

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y" + y = sin x +xcos x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y(4) - 5y" + 4y = eX - xe2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y (5) + 2y(3) + 2y" = 3x2 - I

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x. y(3) - y = eX + 7

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y" - 2y' + 2y = eX sin x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y (5) - y(3) = eX + 2x2 - 5

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y" + 4 y = 3x cos 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y(3) - y" - 1 2y' = x - 2xe-3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y" + 3y' + 2y = x(e-X - e-2x)

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y" - 6y' + 1 3y = xe3x sin 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y(4) + 5y" + 4y = sin x + cos 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y(4) + 9y" = (x2 + I) sin 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. (D - 1 )3 (D2 - 4)y = xeX + e2x + e-2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 21 through 30, set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y(4) - 2y" + y = x2 cos x

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y" + 4y = 2x ; y (O) = I, y'(O) = 2

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y" + 3y' + 2y = eX; y (O) = 0, y'(O) = 3

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y" + 9y = sin 2x ; y (O) = I, y'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y" + y = cos x; y (O) = I, y'(O) = -I

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y" - 2y' + 2y = x + I ; y (O) = 3, y'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y(4) - 4y" = x2; y (O) = y'(O) = I, y"(O) = y(3) (O) = -I

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y(3) - 2y" + y' = I + xex ; y (O) = y'(O) = 0, y"(O) = I

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y" + 2y' + 2y = sin 3x ; y (O) = 2, y'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y(3) + y" = x + e-x; y (O) = I, y'(O) = 0, y"(O) = I

Chapter 2: Problem 2 Elementary Differential Equations 6

Solve the initial value problems in Problems 31 through 40. y(4) - Y = 5 ; y (O) = y'(O) = y" (O) = y(3l (O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Find a particular solution of the equation y(4) _ y(3) _ y" _ y' _ 2y = 8x5 .

Chapter 2: Problem 2 Elementary Differential Equations 6

Find the solution of the initial value problem consisting of the differential equation of Problem 41 and the initial conditions y (O) = y'(O) = y"(O) = y< 3 ) (0) = O.

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) Write cos 3x + i sin 3x = e3ix = (cos x + i sin x)3 by Euler's formula, expand, and equate real and imaginary parts to derive the identities cos3 x = cos x + cos 3x , sin3 x = sin x - sin 3x. (b) Use the result of part (a) to find a general solution of y" + 4 y = cos3 x.

Chapter 2: Problem 2 Elementary Differential Equations 6

Use trigonometric identities to find general solutions of the equations in Problems 44 through 46. y" + y' + y = sin x sin 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

Use trigonometric identities to find general solutions of the equations in Problems 44 through 46. y" + 9y = sin4 x

Chapter 2: Problem 2 Elementary Differential Equations 6

Use trigonometric identities to find general solutions of the equations in Problems 44 through 46. y" + y = X cos3 X

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" + 3y' + 2y = 4eX

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" - 2y' - 8y = 3e2

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" - 4y' + 4y = 2e2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" - 4y = sinh 2x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" + 4y = cos 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" + 9y = sin 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" + 9y = 2 sec 3x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" + y = csc2 x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" + 4y = sin2 x

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 47 through 56, use the method of variation ofparameters to find a particular solution of the given differential equation. y" - 4y = xex

Chapter 2: Problem 2 Elementary Differential Equations 6

You can verify by substitution that Yc = C1 X + C 2 X-1 is a complementary function for the nonhomogeneous secondorder equation But before applying the method of variation of parameters, you must first divide this equation by its leading coefficient x2 to rewrite it in the standard form " I , I Y + -y - - y = 72x3 . x x2 Thus f (x) = 72x3 in Eq. (22). Now proceed to solve the equations in (3 1) and thereby derive the particular solution YP = 3x5

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function Yc are given. Apply the method of Problem 57 to find a particular solution of the equation x2y" - 4xy' + 6y = x3; Yc = C1X2 + C 2 x3

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function Yc are given. Apply the method of Problem 57 to find a particular solution of the equation x 2y" - 3xy' + 4y = X4; Yc = x2(Cl + c 2 ln x)

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function Yc are given. Apply the method of Problem 57 to find a particular solution of the equation 4x2y" - 4xy' + 3y = 8x4/3 ; Yc = C1X + C 2 X3/4

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function Yc are given. Apply the method of Problem 57 to find a particular solution of the equation x2y" + xy' + y = In x; Yc = Cl cos(ln x) + C 2 sin(ln x)

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 58 through 62, a nonhomogeneous second-order linear equation and a complementary function Yc are given. Apply the method of Problem 57 to find a particular solution of the equation (x2 - I)y" - 2xy' + 2y = x2 - I ; Yc = C1X + C 2 ( l + x2)

Chapter 2: Problem 2 Elementary Differential Equations 6

Carry out the solution process indicated in the text to derive the variation of parameters formula in (33) from Eqs. (3 1) and (32).

Chapter 2: Problem 2 Elementary Differential Equations 6

Apply the variation of parameters formula in (33) to find the particular solution YP (x) = -x cos x of the nonhomogeneous equation y" + y = 2 sin x.

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1-4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 2.6.2) its period x" + 9x = l O cos 2t; x(O) = x'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1-4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 2.6.2) its period x" + 4x = 5 sin 3t; x(O) = x'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1-4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 2.6.2) its period x" + 1 00x = 225 cos 5t + 300 sin 5t; x (0) 375, x' (O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1-4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 2.6.2) its period x" + 25x = 90 cos 4t ; x(O) = 0, x'(O) = 90

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1-4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 2.6.2) its period mx" + kx = Fo cos wt with w =1= Wo ; x (0) = xo , x' (0) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems I through 6, express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. In Problems 1-4, graph the solution function x(t) in such a way that you can identify and label (as in Fig. 2.6.2) its period mx" +kx = Fo cos wt with w = Wo ; x(O) = 0, x'(O) = Vo

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 7 through 10, find the steady periodic solution xsp(t) = C cos(wt - a) of the given equation mx" + cx' + kx = F(t) with periodic forcing function F(t) offrequency w. Then graph xsp(t) together with (for comparison) the adjusted forcing function F) (t) = F (t) j mw. x" + 4x' + 4x = IO cos 3t

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 7 through 10, find the steady periodic solution xsp(t) = C cos(wt - a) of the given equation mx" + cx' + kx = F(t) with periodic forcing function F(t) offrequency w. Then graph xsp(t) together with (for comparison) the adjusted forcing function F) (t) = F (t) j mw. x" + 3x' + 5x = -4 cos 5t

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 7 through 10, find the steady periodic solution xsp(t) = C cos(wt - a) of the given equation mx" + cx' + kx = F(t) with periodic forcing function F(t) offrequency w. Then graph xsp(t) together with (for comparison) the adjusted forcing function F) (t) = F (t) j mw. 2X" + 2x' + x = 3 sin l Ot

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 7 through 10, find the steady periodic solution xsp(t) = C cos(wt - a) of the given equation mx" + cx' + kx = F(t) with periodic forcing function F(t) offrequency w. Then graph xsp(t) together with (for comparison) the adjusted forcing function F) (t) = F (t) j mw. x" + 3x' + 3x = 8 cos lOt + 6 sin lOt

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems I I through 14, find and plot both the steady periodic solution xsp(t) = C cos(wt - a) of the given differential equation and the transient solution Xtr(t) that satisfies the given initial conditions. x" + 4x' + 5x = l O cos 3t; x(O) = x'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems I I through 14, find and plot both the steady periodic solution xsp(t) = C cos(wt - a) of the given differential equation and the transient solution Xtr(t) that satisfies the given initial conditions. x" + 6x' + 1 3x = 10 sin 5t; x(O) = x'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems I I through 14, find and plot both the steady periodic solution xsp(t) = C cos(wt - a) of the given differential equation and the transient solution Xtr(t) that satisfies the given initial conditions. x" + 2x' + 26x = 600 cos l Ot; x(O) = 1 0, x'(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems I I through 14, find and plot both the steady periodic solution xsp(t) = C cos(wt - a) of the given differential equation and the transient solution Xtr(t) that satisfies the given initial conditions. x" + 8x' + 25x = 200 cos t + 520 sin t; x(O) = -30, x'(O) = - 10

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 15 through 18 gives the parameters for a forced mass-spring-dashpot system with equation mx" +cx' + kx = Fo cos wt. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C(w) of steady periodic forced oscillations with frequency w. Sketch the graph of C (w) and find the practical resonance frequency w (if any). m = I, c = 2, k = 2, Fo = 2

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 15 through 18 gives the parameters for a forced mass-spring-dashpot system with equation mx" +cx' + kx = Fo cos wt. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C(w) of steady periodic forced oscillations with frequency w. Sketch the graph of C (w) and find the practical resonance frequency w (if any). m=l ,c=4, k = 5, Fo=1 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 15 through 18 gives the parameters for a forced mass-spring-dashpot system with equation mx" +cx' + kx = Fo cos wt. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C(w) of steady periodic forced oscillations with frequency w. Sketch the graph of C (w) and find the practical resonance frequency w (if any). m = l , c = 6, k = 45, Fo = 50

Chapter 2: Problem 2 Elementary Differential Equations 6

Each of Problems 15 through 18 gives the parameters for a forced mass-spring-dashpot system with equation mx" +cx' + kx = Fo cos wt. Investigate the possibility of practical resonance of this system. In particular, find the amplitude C(w) of steady periodic forced oscillations with frequency w. Sketch the graph of C (w) and find the practical resonance frequency w (if any). m = I, c = 1 0, k = 650, Fo = 1 00

Chapter 2: Problem 2 Elementary Differential Equations 6

A mass weighing 1 00 lb (mass m = 3. 1 25 slugs in fps units) is attached to the end of a spring that is stretched I in. by a force of 1 00 lb. A force Fo cos wt acts on the mass. At what frequency (in hertz) will resonance oscillations occur? Neglect damping.

Chapter 2: Problem 2 Elementary Differential Equations 6

A front-loading washing machine is mounted on a thick rubber pad that acts like a spring; the weight W = mg (with g = 9.8 mjs2) of the machine depresses the pad exactly 0.5 cm. When its rotor spins at w radians per second, the rotor exerts a vertical force Fo cos wt newtons on the machine. At what speed (in revolutions per minute) will resonance vibrations occur? Neglect friction.

Chapter 2: Problem 2 Elementary Differential Equations 6

Figure 2.6. 10 shows a mass m on the end of a pendulum (of length L) also attached to a horizontal spring (with constant k). Assume small oscillations of m so that the spring remains essentially horizontal and neglect damping. Find the natural circular frequency Wo of motion of the mass in terms of L, k, m, and the gravitational constant g.

Chapter 2: Problem 2 Elementary Differential Equations 6

A mass m hangs on the end of a cord around a pulley of radius a and moment of inertia I, as shown in Fig. 2.6.11. The rim of the pulley is attached to a spring (with constant k). Assume small oscillations so that the spring remains essentially horizontal and neglect friction. Find the natural circular frequency of the system in terms of m, a, k, I, and g.

Chapter 2: Problem 2 Elementary Differential Equations 6

A building consists of two floors. The first floor is attached rigidly to the ground, and the second floor is of mass m = 1 000 slugs (fps units) and weighs 16 tons (32,000 Ib). The elastic frame of the building behaves as a spring that resists horizontal displacements of the second floor; it requires a horizontal force of 5 tons to displace the second floor a distance of 1 ft. Assume that in an earthquake the ground oscillates horizontally with amplitude Ao and circular frequency w, resulting in an external horizontal force F (t) = m Aow2 sin wt on the second floor. (a) What is the natural frequency (in hertz) of oscillations of the second floor? (b) If the ground undergoes one oscillation every 2.25 s with an amplitude of 3 in., what is the amplitude of the resulting forced oscillations of the second floor?

Chapter 2: Problem 2 Elementary Differential Equations 6

A mass on a spring without damping is acted on by the external force F (t) = Fo cos3 wt. Show that there are two values of w for which resonance occurs, and find both.

Chapter 2: Problem 2 Elementary Differential Equations 6

Derive the steady periodic solution of mx" + CX' + kx = Fo sin wt In particular, show that it is what one would expect-the same as the formula in (20) with the same values of C and w, except with sin(wt - a) in place of cos(wt - a).

Chapter 2: Problem 2 Elementary Differential Equations 6

Given the differential equation mx" + cx' + kx = Eo cos wt + Fo sin wt -with both cosine and sine forcing terms---derive the steady periodic solution / E 2 + F.2 xsp (t) = V 0 0 cos(wt - a - {3), .j(k - m(2)2 + (cw)2 where a is defined in Eq. (22) and {3 = tanI (Fo/Eo). (Suggestion: Add the steady periodic solutions separately corresponding to Eo cos wt and Fo sin wt (see Problem 25).)

Chapter 2: Problem 2 Elementary Differential Equations 6

According to Eq. (2 1 ), the amplitude of forced steady periodic oscillations for the system mx" + cx' + kx = Fo cos wt is given by (a) If c ccr/h, where Ccr = ,J4km, show that C steadily decreases as w increases. (b) If c < ccr/h, show that C attains a maximum value (practical resonance) when

Chapter 2: Problem 2 Elementary Differential Equations 6

As indicated by the cart-with-flywheel example discussed in this section, an unbalanced rotating machine part typically results in a force having amplitude proportional to the square of the frequency w. (a) Show that the amplitude of the steady periodic solution of the differential equation mx" + cx' + kx = mAw2 cos wt (with a forcing term similar to that in Eq. (17 is given by (b) Suppose that c 2 < 2mk. Show that the maximum amplitude occurs at the frequency Wm given by wm = k ( 2mk ) ;;; 2mk - c2 Thus the resonance frequency in this case is larger (in contrast with the result of Problem 27) than the natural frequency Wo = ,Jk/m. (Suggestion: Maximize the square of C.)

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 29 and 30 deal further with the car of Example 5. Its upward displacement function satisfies the equation mx" + cx' + kx = cy' + ky when the shock absorber is connected (so that c > 0). With y = a sin wt for the road surface, this differential equation becomes where Eo = cwa and Fo = ka Apply the result of Problem 26 to show that the amplitude C of the resulting steady periodic oscillation for the car is given by 8' S 0) '0.B9-a 6S-<Because w = 2n v/L when the car is moving with velocity v, this gives C as a function of v.

Chapter 2: Problem 2 Elementary Differential Equations 6

Problems 29 and 30 deal further with the car of Example 5. Its upward displacement function satisfies the equation mx" + cx' + kx = cy' + ky when the shock absorber is connected (so that c > 0). With y = a sin wt for the road surface, this differential equation becomes where Eo = cwa and Fo = ka Figure 2.6. 12 shows the graph of the amplitude function C(w) using the numerical data given in Example 5 (including c = 3000 Ns/m). It indicates that, as the car accelerates gradually from rest, it initially oscillates with amplitude slightly over 5 cm. Maximum resonance vibrations with amplitude about 14 cm occur around 32 mi/h, but then subside to more tolerable levels at high speeds. Verify these graphically based conclusions by analyzing the function C(w). In particular, find the practical resonance frequency and the corresponding amplitude.

Chapter 2: Problem 2 Elementary Differential Equations 6

In the circuit of Fig. 2.7.7, suppose that L = 5 H, R = 25 Q, and the source E of emf is a battery supplying 100 V to the circuit. Suppose also that the switch has been in position 1 for a long time, so that a steady current of 4 A is flowing in the circuit. At time t = 0, the switch is thrown to position 2, so that 1(0) = 4 and E = 0 for t O. Find I (t).

Chapter 2: Problem 2 Elementary Differential Equations 6

Given the same circuit a s i n Problem I, suppose that the switch is initially in position 2, but is thrown to position 1 at time t = 0, so that 1(0) = 0 and E = 100 for t O. Find I (t) and show that I (t) --+ 4 as t --+ +00.

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that the battery in Problem 2 is replaced with an alternating-current generator that supplies a voltage of E(t) = 100 cos 60t volts. With everything else the same, now find I (t).

Chapter 2: Problem 2 Elementary Differential Equations 6

In the circuit of Fig. 2.7.7, with the switch in position I, suppose that L = 2, R = 40, E(t) = 100elOt , and 1(0) = O. Find the maximum current in the circuit for t O.

Chapter 2: Problem 2 Elementary Differential Equations 6

In the circuit of Fig. 2.7.7, with the switch in position I, suppose that E(t) = 100elOt cos60t, R = 20, L = 2, and 1(0) = O. Find l(t).

Chapter 2: Problem 2 Elementary Differential Equations 6

In the circuit of Fig. 2.7.7, with the switch in position I, take L = I, R = 10, and E(t) = 30 cos 60t + 4O sin 60t. (a) Substitute Isp (t) = A cos 60t + B sin 60t and then determine A and B to find the steady-state current Isp in the circuit. (b) Write the solution in the form Isp(t) = C cos(wt - a).

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) Find the charge Q(t) and current I (t) in the RC circuit if E(t) = Eo (a constant voltage supplied by a battery) and the switch is closed at time t = 0, so that Q(O) = O. (b) Show that lim Q(t) = EoC and that lim I (t) = O.

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that in the circuit of Fig. 2.7.8, we have R = 10, C = 0.02, Q(O) = 0, and E(t) = 100e-5t (volts). (a) Find Q(t) and I (t). (b) What is the maximum charge on the capacitor for t 0 and when does it occur?

Chapter 2: Problem 2 Elementary Differential Equations 6

Suppose that in the circuit of Fig. 2.7.8, R = 200, C = 2.5 x 10-4, Q(O) = 0, and E(t) = 100 cos 120t. (a) Find Q(t) and I (t). (b) What is the amplitude of the steady-state current?

Chapter 2: Problem 2 Elementary Differential Equations 6

An emf of voltage E (t) = Eo cos wt is applied to the RC circuit of Fig. 2.7.8 at time t = 0 (with the switch closed), and Q(O) = O. Substitute Q,p(t) = A cos wt + B sin wt in the differential equation to show that the steady periodic charge on the capacitor is EoC Q,p(t) = cos(wt - fJ) .JI + w2R2C2 where fJ = tanI (wRC).

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 11 through 16, the parameters of an RLC circuit with input voltage E(t) are given. Substitute I,p(t) = A cos wt + B sin wt in Eq. (4), using the appropriate value of w, to find the steady periodic current in the form I,p(t) = 10 sin(wt - 8). R = 30 [2, L = 10 H, C = 0.02 F; E(t) = 50 sin 2t V

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 11 through 16, the parameters of an RLC circuit with input voltage E(t) are given. Substitute I,p(t) = A cos wt + B sin wt in Eq. (4), using the appropriate value of w, to find the steady periodic current in the form I,p(t) = 10 sin(wt - 8). R = 200 [2, L = 5 H, C = 0.001 F; E(t) = 100 sin lOt V

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 11 through 16, the parameters of an RLC circuit with input voltage E(t) are given. Substitute I,p(t) = A cos wt + B sin wt in Eq. (4), using the appropriate value of w, to find the steady periodic current in the form I,p(t) = 10 sin(wt - 8). R = 20 [2, L = 10 H, C = om F; E(t) = 200 cos 5t V

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 11 through 16, the parameters of an RLC circuit with input voltage E(t) are given. Substitute I,p(t) = A cos wt + B sin wt in Eq. (4), using the appropriate value of w, to find the steady periodic current in the form I,p(t) = 10 sin(wt - 8). R = 50 [2, L = 5 H, C = 0.005 F; E(t) = 300 cos lOOt + 400 sin lOOt V

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 11 through 16, the parameters of an RLC circuit with input voltage E(t) are given. Substitute I,p(t) = A cos wt + B sin wt in Eq. (4), using the appropriate value of w, to find the steady periodic current in the form I,p(t) = 10 sin(wt - 8). R = 100 [2, L = 2 H, C = 5 X 10-6 F; E(t) = 110 sin 60rrt V

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 11 through 16, the parameters of an RLC circuit with input voltage E(t) are given. Substitute I,p(t) = A cos wt + B sin wt in Eq. (4), using the appropriate value of w, to find the steady periodic current in the form I,p(t) = 10 sin(wt - 8). R = 25 [2, L = 0.2 H, C = 5 X 10-4 F; E(t) = 120 cos 377t

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 17 through 22, an RLC circuit with input voltage E(t) is described. Find the current I (t) using the given initial current (in amperes) and charge on the capacitor (in coulombs). R = 16 [2, L = 2 H, C = 0.02 F; E(t) = 100 V; 1(0) = 0, Q(O) = 5

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 17 through 22, an RLC circuit with input voltage E(t) is described. Find the current I (t) using the given initial current (in amperes) and charge on the capacitor (in coulombs). R = 60 [2, L = 2 H, C = 0.0025 F; E(t) = 100e-t V; 1(0) = 0, Q(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In Problems 17 through 22, an RLC circuit with input voltage E(t) is described. Find the current I (t) using the given initial current (in amperes) and charge on the capacitor (in coulombs). R = 60 [2, L = 2 H, C = 0.0025 F; E(t) = 100elOt V; 1(0) = 0, Q(O) = I

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 20 through 22, plot both the steady periodic current I,p(t) and the total current I (t) = Isp(t) + Itr(t). The circuit and input voltage of Problem II with 1(0) = 0 and Q(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 20 through 22, plot both the steady periodic current I,p(t) and the total current I (t) = Isp(t) + Itr(t). The circuit and input voltage of Problem 13 with 1(0) = 0 and Q(O) = 3

Chapter 2: Problem 2 Elementary Differential Equations 6

In each of Problems 20 through 22, plot both the steady periodic current I,p(t) and the total current I (t) = Isp(t) + Itr(t). The circuit and input voltage of Problem 15 with 1(0) = 0 and Q(O) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider an LC circuit-that is, an RLC circuit with R = O-with input voltage E(t) = Eo sin wt. Show that unbounded oscillations of current occur for a certain resonance frequency; express this frequency in terms of L and C

Chapter 2: Problem 2 Elementary Differential Equations 6

It was stated in the text that, if R, L, and C are positive, then any solution of LI" + RI' + IIC = 0 is a transient solution-it approaches zero as t +00. Prove this.

Chapter 2: Problem 2 Elementary Differential Equations 6

Prove that the amplitude 10 of the steady periodic solution of Eq. (6) is maximal at frequency w = I/.../LC.

Chapter 2: Problem 2 Elementary Differential Equations 6

The eigenvalues in Problems 1 through 5 are all nonnegative. First determine whether A = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y" + AY = 0; y'(0) = 0, y(1) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

The eigenvalues in Problems 1 through 5 are all nonnegative. First determine whether A = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y" + Ay = 0; y'(0) = 0, y'(n) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

The eigenvalues in Problems 1 through 5 are all nonnegative. First determine whether A = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y"+Ay=O; y(-n) = O, y(n) =O

Chapter 2: Problem 2 Elementary Differential Equations 6

The eigenvalues in Problems 1 through 5 are all nonnegative. First determine whether A = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y" + AY = 0; y'(-n) = 0, y'(n) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

The eigenvalues in Problems 1 through 5 are all nonnegative. First determine whether A = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. y" + Ay = 0; y(-2) = 0, y'(2) = 0

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider the eigenvalue problem y" + Ay = 0; y'(O) = 0, y(1) + y'(1) = O. All the eigenvalues are nonnegative, so write A = a2 where a O. (a) Show that A = 0 is not an eigenvalue. (b) Show that y = A cosax + B sin ax satisfies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = l/z. These roots {an I j'" are the abscissas of the points of intersection of the curves y = tan z and y = 1/ z , as indicated in Fig. 2.8. 13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {al\)O and the functions {cos anxlj"', respectively

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider the eigenvalue problem y" + AY = 0; yeO) = 0, y(1) + y'(1) = 0; all its eigenvalues are nonnegative. (a) Show that A = 0 is not an eigenvalue. (b) Show that the eigenfunctions are the functions {sin anxlj"', where an is the nth positive root of the equation tan z = -z . (c) Draw a sketch indicating the roots {an I yo as the points of intersection of the curves y = tan z and y = -z. Deduce from this sketch that an (2n - 1)n /2 when n is large.

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider the eigenvalue problem y" + AY = 0; yeO) = 0, y(l) = y'(I); all its eigenvalues are nonnegative. (a) Show that A = 0 is an eigenvalue with associated eigenfunction Yo(x) = x. (b) Show that the remaining eigenfunctions are given by Yn(x) = sin f3nx, where f3n is the nth positive root of the equation tan z = z . Draw a sketch showing these roots. Deduce from this sketch that f3n (2n + l)n /2 when n is large.

Chapter 2: Problem 2 Elementary Differential Equations 6

Prove that the eigenvalue problem of Example 4 has no negative eigenvalues.

Chapter 2: Problem 2 Elementary Differential Equations 6

Prove that the eigenvalue problem y" + AY = 0; yeO) = 0, y(l) + y'(l) = 0 has no negative eigenvalues. (Suggestion: Show graphically that the only root of the equation tanh z = -z is z = 0.)

Chapter 2: Problem 2 Elementary Differential Equations 6

Use a method similar to that suggested in Problem 10 to show that the eigenvalue problem in Problem 6 has no negative eigenvalues.

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider the eigenvalue problem y" + AY = 0; y(-n) = yen), y'(-n) = y'(n), which is not of the type in (10) because the two endpoint conditions are not "separated" between the two endpoints. (a) Show that AO = 0 is an eigenvalue with associated eigenfunction Yo (x) == 1. (b) Show that there are no negative eigenvalues. (c) Show that the nth positive eigenvalue is n2 and that it has two linearly independent associated eigenfunctions, cos nx and sin nx.

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider the eigenvalue problem y" + 2y' + AY = 0; y eO) = y(1) = O. (a) Show that A = 1 is not an eigenvalue. (b) Show that there is no eigenvalue A such that A < 1. (c) Show that the nth positive eigenvalue is An = n2rr2 + 1, with associated eigenfunction Yn (x) = e -X sin nrr x.

Chapter 2: Problem 2 Elementary Differential Equations 6

Consider the eigenvalue problem y" + 2y' + AY = 0; yeO) = 0, y'(I) = O. Show that the eigenvalues are all positive and that the nth positive eigenvalue is An = a; + 1 with associated eigenfunction Yn (x) = e-x sin anx, where an is the nth positive root of tan z = z .

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) A uniform cantilever beam is fixed at x = 0 and free at its other end, where x = L. Show that its shape is given by w y(x) = __ (x4 -4Lx3 + 6L2x 2 ). 241 (b) Show that y' (x) = 0 only at x = 0, and thus that it follows (why?) that the maximum deflection of the cantilever is Ymax = y(L) = wL 4/(8 I).

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) Suppose that a beam is fixed at its ends x = 0 and x = L. Show that its shape is given by w y(x) = __ (x4 -2Lx3 + L2x 2 ). 241 (b) Show that the roots of y'(x) = 0 are x = 0, x = L, and x = L/2, so it follows (why?) that the maximum deflection of the beam is (L) WL4 Ymax = Y "2 = 3841 ' one-fifth that of a beam with simply supported ends.

Chapter 2: Problem 2 Elementary Differential Equations 6

For the simply supported beam whose deflection curve is given by Eq. (24), show that the only root of y'(x) = in [0, L] is x = Lj2, so it follows (why?) that the maximum deflection is indeed that given in Eq. (25).

Chapter 2: Problem 2 Elementary Differential Equations 6

(a) A beam is fixed at its left end x = 0 but is simply supported at the other end x = L. Show that its deflection curve is (b) Show that its maximum deflection occurs where x = (1 5 -.J33 )L/16 and is about 41 .6% of the maximum deflection that would occur if the beam were simply supported at each end.

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