1550078481-Ordinary_Differential_Equations__Roberts_

182 Ordinary Differential Equations

DEFINITIONS Nonhomogeneous Linear DE and

Associated Homogeneous Linear DE

The general n-th order nonhomogeneous linear differential equa-

tion has the form

where an(x) :j=. 0 in some interval I and b(x) :j=. 0 for some x E J.

The associated homogeneous linear differential equation is

DEFINITION Particular Solution

Any function Yp(x) which satisfies t he nonhomogeneous linear DE (24)

and which contains no arbitrary constant is called a particular solution

of the nonhomogeneous equation.

For example, Yp(x) = 2 is a particular solution of the nonhomogeneous

linear differential equation y" + 4y = 8, since y~ + 4yp = 0 + 4(2) = 8 and Yp

contains no arbitrary constant.

EXAMPLE 9 Verification of a Particular Solution

Show that y(x) = x^2 - 2x is a particular solution of the nonhomogeneous

linear differential equation

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

SOLUTION

Differentiating y(x) = x^2 - 2x three times, we find yC^1 l(x) = 2x - 2,

yC^2 l(x) = 2, and y(^3 ) = 0. Substituting for y(x) and its derivatives in the
DE (26), we see that

yC^3 l - 3yC^2 l + 3y(ll - y = 0 - 3(2) + 3(2x - 2) - (x^2 - 2x)

= -6 + 6x - 6 - x^2 + 2x = - x^2 + 8x - 12.