1550078481-Ordinary_Differential_Equations__Roberts_

218 Ordinary Differential Equations

Since b 2 ( x) = -2e-^3 x is of the form (7) and r = -3 is a root of the auxiliary

equation of order k = 1, a particular solution yp 2 (x) of (12) will have the form

Yp 2 (x) = x^1 e-^3 x(C) = Cxe-^3 x.

Differentiating, we get y~ 2 (x) = C(-3x + l)e-^3 x and y~ 2 (x) = C(9x - 6)e-^3 x.
Substituting into the DE (12), we find C must satisfy

C(9x - 6)e-^3 x + 3C(-3x + l)e-^3 x = -2e-^3 x or - 3Ce-^3 x = -2e-^3 x.

2 2 -3x

Hence, C =

3

and Yp 2 (x) = 3xe.

Adding the particular solutions Yp,(x) and yp 2 (x) of the two nonhomoge-

neous linear differential equations (11) and (12), we find that a particular

solution of the given nonhomogeneous differential equation (10) is

and, consequently, the general solution of (10) is

EXERCISES 4.4

Determine the general solution of the following nonhomogeneous

linear differential equations.

1. y(^4 ) - 6y(^3 ) + 13y(^2 ) - 12y(ll + 4y = 2ex - 4e^2 x

2. y(^4 ) + 4y(^2 ) = 24x^2 - 6x + 14 + 32 cos 2x


3. y(^4 ) + 2y(^2 ) + y = 3 + cos2x

4. y(^4 ) - 3y(^3 ) + 3y(^2 ) - y(ll = 6x - 20 - 120x^2 ex

5. y(^3 ) - 6y(^2 ) + 21y(ll - 26y = 36e^2 x sin3x


6. y(^3 ) + y(^2 ) - y(l) - y = (2x^2 + 4x + 8) cosx + (6x^2 + 8x + 12) sinx


7. y(^6 ) - 12y(^5 ) + 63y(^4 ) - 184y(^3 ) + 315y(^2 ) - 300y(ll + 125y =

ex(48cosx + 96sinx)