1550078481-Ordinary_Differential_Equations__Roberts_

216 Ordinary Differential Equations

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

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

the form

where Am, Am-l, ... , A 1 , and Ao are unknown constants to be determined.

Case 3. If r = a+ bi and r = a - bi are roots of multiplicity k (k 2: 0)

of the auxiliary equation of the associated homogeneous linear differential

equation (2) and if b(x) has the form

(8) b(x) = eax (bmxm + bm-1Xm-l + · · · + b1x + bo) sin bx

or the form

then there is a particular solution of (1) of the form

Yp = xkeax[(Amxm + Am-1Xm-l + · · · + A1x +Ao) cos bx+

(Bmxm + Bm-1Xm-l + · · · + B1x +Bo) sin bx]

where Am, Am-1, .· .. , Ai, Ao, Bm, Bm-1, ... , Bi, and Bo are unknown

constants which are to be determined.

When b(x) is a sum of terms of the form (6), (7), (8), or (9)- that is, when
b(x) = bi(x) + bz(x) + · · · + b 8 (x) where b 1 (x), bz(x), ... , b 8 (x) all have
one of the forms (6), (7), (8), or (9)- it is usually easier, first, to calculate
s particular solutions, Yp;(x), 1:::; j:::; s, to the corresponding s separate
nonhomogeneous linear differential equations

And then add the s particular solutions to obtain Yp(x) = Yp 1 (x) +yp 2 (x) +
· · · + YPs (x) which is a particular solution of the original differential equation

+ · · ·+bs(x).

The following example illustrates this procedure.