c04.qxd 6/18/11 3:06 PM Page 64 CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods Major Changes This chapter was completely rewritten in the eighth 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 ϫ 2 matrices. Accordingly, Chap. 4 makes modest use of 2 ϫ 2 matrices. n ϫ 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 ϫ 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 ϫ 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 ϫ 2 matrices), consisting first of the determination of the eigenvalues l1, l2 (not necessarily numerically distinct) as solutions of the characteristic equation, that is, the quadratic equation a11 Ϫ l a12 ϭ (a11 Ϫ l)(a22 Ϫ l) Ϫ a12a21 a21 a22 Ϫ l 2 ϭ l Ϫ (a11 ϩ a22)l ϩ a11a22 Ϫ a12a21 ϭ 0, 2 2 and then an eigenvector corresponding to l1 with components x 1, x 2 from (a11 Ϫ l1)x 1 ϩ a12x 2 ϭ 0 and an eigenvector corresponding to l2 from (a11 Ϫ l2)x 1 ϩ a12x 2 ϭ 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. 64 c04.qxd 6/18/11 3:06 PM Page 65 Instructor’s Manual 65 SECTION 4.0. For Reference: 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. Main Content Matrices, vectors Algebraic matrix operations Differentiation of vectors Eigenvalue problems for 2 ϫ 2 matrices Important Concepts and Facts Matrix, column vector and row vector, multiplication Linear independence Eigenvalue, eigenvector, characteristic equation Some Details in Content Most of the material is explained in terms of 2 ϫ 2 matrices, which play the major role in Chap. 4; indeed, n ϫ n matrices for general n occur only briefly in Sec. 4.2 and at the beginning in Sec. 4.3. Hence the demand of linear algebra on the student in Chap. 4 will be very modest, and Sec. 4.0 is written accordingly. In particular, eigenvalue problems lead to quadratic equations only, so that nothing needs to be said about difficulties encountered with 3 ϫ 3 or larger matrices. Example 1. Although the later sections include many eigenvalue problems, the complete solution of such a problem (the determination of the eigenvalues and corresponding eigenvectors) is given in Sec. 4.0. Emphasize to your students that the eigenvalues of a given square matrix are uniquely determined (and some of them can very well be 0), whereas eigenvectors must not be zero vectors and are determined only up to a nonzero multiplicative constant. SECTION 4.1. Systems of ODEs as Models in Engineering Applications, page 130 Purpose. In this section the student will gain a first impression of the importance of systems of ODEs in physics and engineering and will learn why they occur and why they lead to eigenvalue problems. Main Content Mixing problem Electrical network Conversion of single equations to system (Theorem 1) The possibility of switching back and forth between systems and single ODEs is practically quite important because, depending on the situation, the system or the single ODE will be the better source for obtaining the information sought in a specific case. Background Material. Secs. 2.4, 2.8. Short Courses. Take a quick look at Sec. 4.1, skip Sec. 4.2 and the beginning of Sec. 4.3, and proceed directly to solution methods in terms of the examples in Sec. 4.3. c04.qxd 6/18/11 3:06 PM Page 66 66 Instructor’s Manual Some Details on Content Example 1 extends the physical system in Sec. 1.3, consisting of a single tank, to a system of two tanks. The principle of modeling remains the same. The problem leads to a typical eigenvalue problem, and the solutions show typical exponential increases and decreases to a constant value. Problem Set 4.1 The mixing problems (Probs. 1–6) should lead to an understanding of the physical parameters involved (tank size, flow rate, amount of fertilizer), similarly in the networks in Probs. 7–9. Problems 10–13 show conversions from ODEs to first-order systems. Problem 14 shows the principle of extending the physical system in Sec. 2.5 to a system of more than one mass and spring, with (11) probably best understood by looking at Fig. 81. SOLUTIONS TO PROBLEM SET 4.1, page 136 2. The system is y1r ϭ 0.02y2 Ϫ 0.01y1 yr2 ϭ 0.01y1 Ϫ 0.02y2 where 0.01 appears because we divide by the content of the tank T1, which is twice the old value. In proper order, the system becomes y1r ϭϪ0.01y1 ϩ 0.02y2 yr2 ϭ 0.01y1 Ϫ 0.02y2. As a single vector equation, Ϫ0.01 0.02 yr ϭ Ay, where A ϭ . c 0.01 Ϫ0.02 d A has the eigenvalues l1 ϭ 0 and l2 ϭϪ0.03 and corresponding eigenvectors 1 1 x(1) ϭ , x(2) ϭ , c 0.5 d c Ϫ1 d respectively. The corresponding general solution is ؊0.03t (2) (1) y ϭ c1x ϩ c2x e . From the initial values, 1 1 0 y(0) ϭ c1 ϩ c2 ϭ . c 0.5 d c Ϫ1 d c 150 d In components this is c1 ϩ c2 ϭ 0, 0.5c1 Ϫ c2 ϭ 150. Hence c1 ϭ 100, c2 ϭϪ100. This gives the solution 1 1 .y ϭ 100 Ϫ 100 e؊0.03t c 0.5 d c Ϫ1 d In components, ؊0.03t y1 ϭ 100(1 Ϫ e ) ϭ 1 ϩ ؊0.03t y2 100(2 e ). c04.qxd 6/18/11 3:06 PM Page 67 Instructor’s Manual 67 4. With Flow rate a ϭ Tank size we can write the system that models the process in the following form: y1r ϭ ay2 Ϫ ay1 yr2 ϭ ay1 Ϫ ay2, ordered as needed for the proper vector form y1r ϭϪay1 ϩ ay2 yr2 ϭ ay1 Ϫ ay2. In vector form, Ϫaa yr ϭ Ay, whereA ϭ . c a Ϫa d The characteristic equation is (l ϩ a)2 Ϫ a2 ϭ l2 ϩ 2al ϭ 0. Hence the eigenvalues are 0 and Ϫ2a. Corresponding eigenvectors are 1 1 and c 1 d c Ϫ1 d respectively. The corresponding “general solution” is 1 1 ؊2at y ϭ c1 ϩ c2 e . c 1 d c Ϫ1 d This result is interesting. It shows that the solution depends only on the ratio a, not on the tank size or the flow rate alone. Furthermore, the larger a is, the more rapidly y1 and y2 approach their limit. The term “general solution” is in quotation marks because this term has not yet been defined formally, although it is clear what is meant. 6. The matrix of the system is Ϫ0.02 0.02 0 A ϭ 0.02 Ϫ0.04 0.02 . 0 0.02 Ϫ0.02 The characteristic polynomial isD T l3 ϩ 0.08l2 ϩ 0.0012l ϭ l(l ϩ 0.02)(l ϩ 0.06). This gives the eigenvalues and corresponding eigenvectors 1 1 1 (1) (2) (3) l1 ϭ 0, x ϭ 1 , l2 ϭϪ0.02, x ϭ 0 , l3 ϭϪ0.06, x ϭ Ϫ2 . 1 Ϫ1 1 D T D T D T c04.qxd 6/18/11 3:06 PM Page 68 68 Instructor’s Manual Hence a “general solution” is 1 1 1 ؊0.02t ؊0.06t ϭ ϩ ϩ Ϫ y c1 1 c2 0 e c3 2 e . 1 Ϫ1 1 We use quotation marksD sinceT theD conceptT of a generalD solutionT has not yet been defined formally, although it is clear what is meant.