Page 320 - Advanced engineering mathematics

300 CHAPTER 10 Systems of Linear Differential Equations

Then
(t) = c 1 1 (t) + ··· + c n n (t)

for all t in I, because now (t) and c 1 1 (t)+···+c n n (t) are both solutions of the initial value
problem

X = AX;X(t 0 ) = (t 0 )
and this solution is unique. This shows that any solution (t) of the system X = AX is a linear

combination of 1 (t),··· , n (t).

We call

c 1 1 (t) + ··· + c n n (t)
the general solution of X =AX when these solutions are linearly independent. Every solu-

tion is contained in this expression by varying the choices of the constants. In the language

of linear algebra, the set of all solutions of X =AX is a vector space of dimension n, hence
any n linearly independent solutions form a basis.

EXAMPLE 10.3
We have seen that
3t 3t
−2e (1 − 2t)e
1 (t) = 3t and 2 (t) = 3t
e te
are linearly independent solutions of
1 −4

X = X.
1 5
The general solution is
X(t) = c 1 1 (t) + c 2 2 (t).

We know the general solution of X = AX if we have n linearly independent solutions.

These solutions are n × 1 matrices. We can form an n × n matrix using these n solutions
as columns. Such a matrix is called a fundamental matrix for the system. In terms of this
fundamental matrix, we can write the general solution in the compact form
c 1 1 + c 2 2 + ··· + c n n = C.

EXAMPLE 10.4
Continuing Example 10.3, form a 2 × 2 matrix using the linearly independent solutions 1 (t)
and 2 (t) as columns:

3t 3t
−2e (1 − 2t)e
(t) = .
e 3t te 3t

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-300 27410_10_ch10_p295-342