Page 321 - Advanced engineering mathematics

10.1 Linear Systems 301

(t) is a fundamental matrix for this system. The general solution c 1 1 + c 2 2 can be written
as C:
3t 3t
−2e + (1 − 2t)e c 1
C = 3t 3t
e + te c 2
3t 3t 3t 3t
c 1 (−2e ) c 2 (1 − 2t)e −2e (1 − 2t)e
= 3t 3t = c 1 3t + c 2 3t
c 1 e c 2 te e te
= c 1 1 (t) + c 2 2 (t).
In an initial value problem, x 1 (t 0 ),··· , x n (t 0 ) are given. This information specifies the n × 1
matrix X(t 0 ). We usually solve an initial value problem by finding the general solution of the
system and then solving for the constants to find the particular solution satisfying the initial
conditions. It is often convenient to use a fundamental matrix to carry out this plan.

EXAMPLE 10.5

Solve the initial value problem

1 −4 −2

X = X;X(0) = .
1 5 3
The general solution is C, with the fundamental matrix of Example 10.4. To solve the initial
value problem we must choose C so that

−2
X(0) = (0)C = .
3
This is the algebraic system

−2 1 −2
C = .
1 0 3
The solution for C is
−1
−2 1 −2 01 −2 3
C = = = .
1 0 3 12 3 4
The unique solution of the initial value problem is
3t 3t
3 −2e − 8te
X(t) = (t) = 3t 3t .
4 3e + 4te
10.1.2 The Nonhomogeneous System

We will develop an analog of Theorem 2.5 for the nonhomogeneous linear system X =AX + G.

The key observation is that, if 1 and 2 are any two solutions of this nonhomogeneous system,

then their difference 1 − 2 is a solution of the homogeneous system X = AX. Therefore, if
is a fundamental matrix for this homogeneous system, then
1 − 2 = K
for some constant n × 1matrix K, hence
1 = 2 + K.

We will state this result as a general theorem.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 20:32 THM/NEIL Page-301 27410_10_ch10_p295-342