1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 217

EXAMPLE 4 General Solution of a Nonhomogeneous Differential

Equation by Superposition

Find the general solution of

(10)

SOLUTION

y" + 3y' = 4x^2 - 2e-^3 x.

The associated homogeneous equation is y" + 3y' = 0. And the auxiliary
equation r^2 + 3r = 0 has roots 0 and -3. So the complementary solution of
the DE (10) is

Yc(x) = C1 + c2e-^3 x.

Next, we find two particular solutions yp, (x) and yp 2 (x ) to the two nonho-
mogeneous differential equations:

(11) y" + 3y' = 4x^2 = b1 ( x)

(12) y" + 3y' = -2e-^3 x = b2(x).

Since b 1 ( x) 4x^2 is of the form ( 6) and r = 0 is a root of the auxiliary

equation of order k = 1, we seek a particular solution yp, (x) of the DE (11)

of the form

Yp, (x) = x^1 (A2x^2 + A1x +Ao)= A2x^3 + A1x^2 + Aox.

Differentiating, we find y~ 1 (x) = 3A2x^2 +2A 1 x+Ao and y~ 1 (x) = 6A2x+2A1.
Substituting into the DE (11), we see A2, A 1 , and Ao must satisfy

(6A2x + 2A1) + 3(3A2x^2 + 2A1x +Ao) = 4x^2

or
9A2x^2 + (6A2 + 6A1)x + (2A1 + 3Ao) = 4x^2.

Equating coefficients of x^2 , x, and x^0 = 1, we see A2, A 1 , and Ao must satisfy
the following system of three linear equations simultaneously.

9A2 = 4

6A2 + 6A1=0

2A1+3Ao = 0.
4 -4 8
Solving this system, we find A 2 = g' A 1 = g' and Ao=
27

. Consequently,

4 3 4 2 8

YP1 (x ) = gx - g x + 27x.