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10.3 Solution of X = AX + G 313

this as a template, look for a particular solution of the nonhomogeneous system of the form
p (t) = (t)U(t), where U(t) is an n × 1 matrix to be determined.
Substitute this proposed particular solution into the nonhomogeneous system to obtain

( U) = U + U = A( U) + G = (A )U + G. (10.3)

is a fundamental matrix for the homogeneous system, so = A . Therefore, U = (A )U

and equation (10.3) becomes
U = G.

Since is nonsingular,
−1
U = G.

Then

−1
U(t) = (t)G(t)dt
in which we integrate a matrix by integrating each element of the matrix. Once we have U(t),we
have the general solution
X(t) = (t)C + (t)U(t)
of the nonhomogeneous system.

EXAMPLE 10.12

We will solve the system

1 −10 t

X = X + .
−1 4 1
The eigenvalues of A are −1,6 with corresponding eigenvectors

5 −2
and .
1 1
A fundamental matrix for X = AX is

−t 6t
5e −2e
(t) = −t 6t .
e e
Compute
1 e t 2e t
−1
(t) = −6t −6t .
7 −e 5e
This inverse is most easily computed using MAPLE. In this 2 × 2 case we could also proceed as
in Example 7.28 of Section 7.7.
With this inverse matrix, we have
1 e t 2e t
t
−1
U (t) = (t)G(t) = −6t −6t
7 −e 5e 1
1 2e + te t
t
= .
7 5e −6t − te −6t
Then

−1
U(t) = (t)G(t)dt
t
(t + 1)e /7
= −6t −6t .
(−29/252)e + (1/42)te

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October 14, 2010 20:32 THM/NEIL Page-313 27410_10_ch10_p295-342