Page 336 - Advanced engineering mathematics

316 CHAPTER 10 Systems of Linear Differential Equations

10.4 Exponential Matrix Solutions

ax
A differential equation y = ay with a as a constant has the general solution y = ce . This leads

us to ask whether there is an analogous solution for the system X = AX with A as an n × n real

constant matrix.
Recall that
1 1
ax 2 3
e = 1 + ax + (ax) + (ax) + ··· .
2 3!
At
Define the exponential matrix e by
1 1
At 2 2 3 3
e = I n + At + A t + A t + ··· ,
2 3!
whenever the infinite series defining the i, j element on the right converges for i and j
varying from 1 through n.

It is routine to verify that
At Bt
e (A+B)t = e e
if A and B are n × n real matrices that commute, that is, if
AB = BA.
Differentiate a matrix by differentiating each element of the matrix. Using the fact that A is a
constant matrix with derivative zero (the n × n zero matrix), we obtain from the definition that
At
At
(e ) = Ae ,
which has the same form as the familiar
at
(e ) = ae .
at
This derivative formula leads to the main point.

THEOREM 10.8

At
Let A be an n × n real, constant matrix and K be any n × 1 matrix of constants. Then e K is a
solution of X = AX. In particular, e is a fundamental matrix for this system.
At

At
The proof is immediate by differentiating. Upon setting X(t) = e K,wehave
d
At
At
e K = Ae K = AX.
X (t) =
dt
We therefore have the general solution of X = AX if we can compute the exponential

At
matrix e . Except for very simple cases this is impractical by hand and requires a computational
software package. If MAPLE is used, the command
exponential(A,t)
At
will return e if A has been defined and n is not “too large.”

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-316 27410_10_ch10_p295-342