1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 183

Since y(x) satisfies the DE (26) and contains no arbitrary constant, y(x) is a
particular solution of (26).

REPRESENTATION THEOREM FOR N-TH ORDER NON-

HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS

If Yp ( x) is any particular solution on the interval I of the nonhomogeneous

linear differential equation

(24)

an(x)y(n)(x) + an-1(x)y(n-l)(x) + · · · + a1(x)yCll(x) + ao(x)y(x) = b(x),

and ify 1 (x),y2(x), ... ,yn(x) are n linearly independent solutions on I of

the associated homogeneous equation

then every solution of the DE (24) on the interval I has the form

where c 1 , c 2 , ... , Cn are suitably chosen constants.

Proof: Let Yc(x) = C1Y1(x) + c2y2(x) + · · · + CnYn(x) where the Ci a re

arbitrary constants and let z(x) be any solution of the nonhomogeneous

linear DE (24) on I. In order to prove this theorem, we must show that

it is possible to choose the Ci so that z(x) = Yc(x ) + yp(x). Since z(x)

and Yp(x) are both solutions on the interval I of the nonhomogeneous

DE (24), w(x) = z(x) - yp(x) is a solut ion on I of the associated homoge-

neous DE (25). By the representation t heorem for n-th order homogeneous

linear differential equations there exist constants c1, c2, ... , Cn such that

Hence, z(x) = Yc(x) + yp(x) for suitably chosen constants c 1 , c2, ... , Cn.

DEFINITIONS Complementary Solution and General Solution

Let y 1 (x),y 2 (x),... , yn(x) be n linearly independent solutions on I of

the homogeneous DE (25) associated with the nonhomogeneous DE (24).

The linear combination Ye ( x) = C1 Y1 ( x ) + c2y2 ( x ) + · · · + CnYn ( x) wh ere

c 1 , c 2 , ... , Cn are arbitrary constants is call ed the complementary solution