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CHAPTER 4
Systems of ODEs. Phase Plane.
Qualitative Methods
Major Changes
This chapter was completely rewritten in the previous edition, on the basis of suggestions
by instructors who have taught from it and my own recent experience. The main reason
for rewriting was the increasing emphasis on
linear algebra
in our standard curricula, so
that we can expect that students taking material from Chap. 4 have at least some working
knowledge of 2
3
2 matrices.
Accordingly, Chap. 4 makes modest use of 2
3
2 matrices.
n
3
n
matrices are mentioned
only in passing and are immediately followed by illustrative examples of systems of two
ODEs in two unknown functions, involving 2
3
2 matrices only. Section 4.2 and the
beginning of Sec. 4.3 are intended to give the student the impression that for firstorder
systems, one can develop a theory that is conceptually and structurally similar to that in
Chap. 2 for a single ODE. Hence if the instructor feels that the class may be disturbed by
n
3
n
matrices, omission of the latter and explanation of the material in terms of two
ODEs in two unknown functions will entail no disadvantage and will leave no gaps of
understanding or skill.
To be completely on the safe side, Sec. 4.0 is included for reference, so that the
student will have no need to search through Chap. 7 or 8 for a concept or fact needed in
Chap. 4.
Basic throughout Chap. 4 is the
eigenvalue problem
(for 2
3
2 matrices), consisting
first of the determination of the eigenvalues
l
1
,
2
(not necessarily numerically distinct)
as solutions of the characteristic equation, that is, the quadratic equation
jj
5
(
a
11
2
)(
a
22
2
)
2
a
12
a
21
5
2
2
(
a
11
1
a
22
)
1
a
11
a
22
2
a
12
a
21
5
0,
and then an eigenvector corresponding to
1
with components
x
1
,
x
2
from
(
a
11
2
1
)
x
1
1
a
12
x
2
5
0
and an eigenvector corresponding to
2
from
(
a
11
2
2
)
x
1
1
a
12
x
2
5
0.
It may be useful to emphasize early that eigenvectors are determined only up to a nonzero
factor and that in the present context, normalization (to obtain unit vectors) is hardly of
any advantage.
If there are students in the class who have not seen eigenvalues before (although the
elementary theory of these problems does occur in every uptodate introductory text on
beginning linear algebra), they should not have difficulties in readily grasping the meaning
of these problems and their role in this chapter, simply because of the numerous examples
and applications in Sec. 4.3 and in later sections.
Section 4.5 includes three famous applications, namely, the
pendulum
and
van der
Pol equations
and the
Lotka–Volterra predator–prey population model.
a
12
a
22
2
a
11
2
a
21
67
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9/21/05
11:08 AM
Page 67
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View Full DocumentSECTION 4.0. Basics of Matrices and Vectors, page 124
Purpose.
This section is for reference and review only, the material being restricted to
what is actually needed in this chapter, to make it selfcontained.
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 Spring '08
 KABA

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