1550078481-Ordinary_Differential_Equations__Roberts_

184 Ordinary Differential Equations

of the nonhomogeneous DE (24). Observe that the complementary solution

is the general solution of the associated homogeneous equation (25).

The general solution of the nonhomogeneous DE (24) is y(x) = Yc(x) +

yp(x) where Yc(x) is the complementary solution and yp(x) is any particular

solution.

EXAMPLE 10 General Solution of a Third-Order,

Nonhomogeneous Linear Differential Equation

Write the general solution of the nonhomogeneous linear differential equa-
tion

(26) yC^3 ) - 3y(^2 ) + 3y(l) - y = -x^2 + 8x - 12.

SOLUTION
In example 9, we showed that a particular solution of the DE (26) is
yp(x) = x^2 - 2x. And in example 8, we showed that the general solution

of the associated homogeneous equation yC^3 ) - 3y(^2 ) + 3y(l) - y = 0 is

Yc(x) = c1ex + c2xex + c3x^2 ex where c1,c2, and c3 are arbitrary constants.
Hence, the general solution of the nonhomogeneous DE (26) is

y( x ) = Yc(x) + Yp(x) = c1ex + c2xex + c3x^2 ex + x^2 - 2x,

where c1, c2, and c3 are arbitrary constants.

In order to solve a nonhomogeneous, linear differential equation, we need
to find a complementary solution- which is a linear combination of n
linearly independent solutions of the associated homogeneous equation- and
a particular solution. In t his chapter, we will show how to solve a nonhomoge-
neous linear differential equation on the interval ( -oo, oo) when the functions
an(x), an-1(x), ... , ao(x) are all constants. In chapter 7, we will show how to
write the general n-th order nonhomogeneous linear differential equation ( 1)
as a system of n first-order differential equations. Then we will be able to gen-
erate a numerical solution to the initial value problem consisting of the initial
conditions (2), and the linear nonhomogeneous differential equation (1) for
arbitrary functions an(x), an-i(x), ... , ao(x), b(x).