MA441 - CH04 - im04.qxd 9/21/05 11:08 AM Page 67 CHAPTER 4...

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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 first-order 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 up-to-date 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 im04.qxd 9/21/05 11:08 AM Page 67
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SECTION 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 self-contained.
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MA441 - CH04 - im04.qxd 9/21/05 11:08 AM Page 67 CHAPTER 4...

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