1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 215

Substituting yp(x ) and its derivatives into equation (5), we find B must satisfy

B(4x + 4)e^2 x - 3(B(2x + l)e^2 x) + 2(Bxe^2 x) = 4e^2 x or Be^2 x = 4e^2 x.

So B = 4, and a particular solut ion of (5) is yp(x) = 4 xe^2 x. Therefore, the

general so lution of (5) is y( x ) = c1ex + c2e^2 x + 4xe^2 x.

From the previous three examples, we have discovered that the form of the
particular solution of a nonhomogeneous differential equation with constant
coefficients depends upon the roots of the auxiliary equation of the associated
homogeneous differential equation as well as the nonhomogeneity itself. The
following three cases delineate this relationship more explicitly.

FORM OF THE PARTICULAR SOLUTION OF A

NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATION

WITH CONSTANT COEFFICIENTS

BASED ON THE ROOTS OF AUXILIARY EQUATION

AND THE FORM OF THE NONHOMOGENEITY

Case 1. If r = 0 is a root of multiplicity k of the auxiliary equation of the

associated homogeneous linear differential equation (2) where k ::;:: 0 (Here

k = 0 corresponds to r = 0 not being a root of the auxiliary equ ation.) and

if b( x) is a polynomial of degree m- that is , if

(6)

where bm, bm-l, ... , b 1 , and b 0 are constants and bm =J=. 0, then there is a

particular solution of the nonhomogeneous linear differential equation (1) of

the form

Yp = xk(Amxm + Am-1Xm-l + · · · + A1x +Ao)

where Am, Am_ 1 , ... , A 1 , and Ao are unknown constants which are to be

determined so that YP satisfies (1). Determination of the actual values of

these constants requires n differentiations of yp, substitution of YP and its

n derivatives into (1), and then the solution of a resulting system of linear

equ ations in them+ 1 unknowns Am, Am-1, ... , A1, Ao.

Case 2. If r = a is a root of mult iplicity k (k ::;:: 0) of the auxiliary

equation of the associated homogeneous linear differential equation (2) and

if b(x) has the form

(7)