Page 338 - Advanced engineering mathematics

318 CHAPTER 10 Systems of Linear Differential Equations

Now
t

p (t) = (t − s)G(s)ds = (t) ∗ G(t).
0
If we take the Laplace transform of a matrix by applying the transform to each element, then the
last equation is a convolution formula for a particular solution.
To illustrate the idea, consider the system
2t
1 −4 e

X = X + .
1 5 t
Compute
3t 3t
(1 − 2t)e −4te
At
(t) = e = 3t 3t .
te (1 + 2t)e
A particular solution of the system is
t
p (t) = (t − s)G(s)ds
0
t 3(t−s) 3(t−s) 2s

(1 − 2(t − s))e −4(t − s)e e
= 3(t−s) 3(t−s) ds
(t − s)e (1 + 2(t − s))e s
0
t 3t −s 3t −3s

(1 − 2t + 2s)e e − 4s(t − s)e e
= 3t −s 3t −3s ds
(t − s)e e + s(1 + 2t − 2s)e e
0
t 3t −s 3t −3s
[(1 − 2t + 2s)e e − 4s(t − s)e e ]ds
= 0 t
[(t − s)e e + (1 + 2t − 2s)e e ]ds
3t −3s
3t −s
0
2t 89 3t 22 3t 4 8
−3e + e − te − t −
9
= 2t 11 27 3t 28 3t 1 9 1 27 .
e + te − e − t +
9 27 9 27
The general solution is X(t) = (t)C + p (t), in which C is an n × 1 matrix of constants.
SECTION 10.4 PROBLEMS
In each of the following, use a software package to com- n × n diagonal matrix having e d j t as its jth diagonal
At
pute e , obtaining a fundamental matrix for the system element.
X = AX,.

7. Let A be an n × n matrix of numbers, and let P be an
−1
n ×n nonsingular matrix of numbers. Let B=P AP.

−1 1
1. A = Show that
−5 1
−1 At
Bt
e = P e P.

−2 1
2. A = From this, conclude that
2 −1
Bt
−1
At
e = Pe P .
5 −2
3. A =
4 8 8. Use the results of Problems 6 and 7 to show that, if
−1
P diagonalizes A,so P AP = D, which is a diagonal
4 −1
4. A = matrix with diagonal elements d j .Then
2 −2
At
Dt
−1
e = Pe P ,
1 0 1
⎛ ⎞
5. A = ⎝ −2 1 1 ⎠ where e Dt is the diagonal matrix having e d j t as main
1 −1 0 diagonal elements.
6. Let D be an n × n diagonal matrix of numbers, 9. Use the result of Problem 8 to determine the exponen-
with jth diagonal element d j . Show that e Dt is the tial matrix in each of Problems 1 and 2.
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October 14, 2010 20:32 THM/NEIL Page-318 27410_10_ch10_p295-342