Page 352 - Advanced engineering mathematics

332 CHAPTER 10 Systems of Linear Differential Equations

y

3

2

1

x
–3 –2 –1 0 1 2 3 4
–1

–2

FIGURE 10.16 Phase portrait in Example
10.19.

EXAMPLE 10.20
The system

33

X = X
15
has a nodal source at the origin because the eigenvalues are 2 and 6 and these are positive and
distinct. The general solution is

−3 2t 1 6t
X(t) = c 1 e + c 2 e
1 1
and a phase portrait is shown in Figure 10.17.

Case 2: The Eigenvalues of A are of Opposite Sign
Suppose the eigenvalues are μ and λ, with μ< 0 <λ. The general solution will again have the
form

y
2.5
2

1.5
1

0.5
x
–2 –1.5 –1 –0.5 0
–0.5

–1

FIGURE 10.17 Phase portrait in Example
10.20.

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-332 27410_10_ch10_p295-342