1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 393

EXAMPLE 4 Solving the Nonhomogeneous

Initial Value Problem (12)

For HR = 2 and ER = 3, solve the nonhomogeneous initial value prob-

lem (12).

SOLUTION

Replacing HR by 2 and ER by 3 in system (12), we see we must solve the

initial value problem

(13a)

(13b) u(O) = 0.

In order to solve the nonhomogeneous initial value problem (13), we must first
find the general solution, Uc, of the associated homogeneous linear system
u' = Au. We will do this with the aid of our computer software. Next,
we must find a particular solution, up, of t he nonhomogeneous system (13a)
u' = Au+ b. The general solution of (13a) is then u = Uc+ up. Finally,
we determine values for the constants appearing in the general solution which
will satisfy the initial conditions (13b).
We ran the computer program EI GEN and found the eigenvalues of A to be
0, 0, and ±2i. We also found the associated linearly independent eigenvectors
to be

So the complementary solution- that is, the real general solution of t he asso-

ciated homogeneous equation u' =Au- is

(14) Uc = C1 v 1 + c2v2 + c3 [ (cos 2t)r - (sin 2t)s] + c4[(sin 2t)r + (cos 2t)s].

We now use the method of undetermined coefficients to find a particular
solution of the nonhomogeneous system of differential equations (13a). We
used this method in chapter 4 to find particular solutions of n-th order linear