1550078481-Ordinary_Differential_Equations__Roberts_

394 Ordinary Differential Equations

nonhomogeneous differential equations with constant coefficients. Recall that
the method consisted of judiciously guessing the form of the particular solu-
tion with coefficients unknown and then determining specific values for the
unknown coefficients so as to satisfy the differential equation. The method
of undetermined coefficients works only when the matrix A is constant and
the components of the nonhomogeneity b are polynomials, exponential or
sinusoidal functions, or the sums and products of these functions.

Since the vector b of system (13a) is a constant, we attempt to find a
particular solution, up, which is also a constant. Hence, we assume

where a, /3, /, and o are unknown constants. Differentiating, we find u~ = 0.

Substituting for U p and u~ in (13a), we see that

u~ = Aup + b or 0 = Aup + b or Aup = - b.

Computing Aup and setting the result equal to -b, we find a, /3, /, and o

must simultaneously satisfy

/3 = 0

-20 = -3

0=0

2/3 = 0

This is impossible, since we must have both o = 3/2 and o = 0. Thus, there

is no particular solution of the form assumed. That is, no constant vector is
a particular solution of (13a).

Having failed to find a constant particular solution of (13a), we next seek
a particular solution in which each component is a linear function. Thus, we
assume u p has the form

(

a+ bt)

c+ dt

U p= e +ft

g+ ht

where a, b, c, d, e, f, g, and h are unknown constants. Differentiating, we

have