MA441 - CH03 - im03.qxd 9/21/05 11:04 AM Page 59 CHAPTER 3...

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CHAPTER 3 Higher Order Linear ODEs This chapter is new. Its material is a rearranged and somewhat extended version of material previously contained in some of the sections of Chap 2. The rearrangement is such that the presentation parallels that in Chap. 2 for second-order ODEs, to facilitate comparisons. Root Finding For higher order ODEs you may need Newton’s method or some other method from Sec. 19.2 (which is independent of other sections in numerics) in work on a calculator or with your CAS (which may give you a root-finding method directly). Linear Algebra The typical student may have taken an elementary linear algebra course simultaneously with a course on calculus and will know much more than is needed in Chaps. 2 and 3. Thus Chaps. 7 and 8 need not be taken before Chap. 3. In particular, although the Wronskian becomes useful in Chap. 3 (whereas for n 5 2 one hardly needs it), a very modest knowledge of determinants will suffice. (For n 5 2 and 3, determinants are treated in a reference section, Sec. 7.6.) SECTION 3.1. Homogeneous Linear ODEs, page 105 Purpose. Extension of the basic concepts and theory in Secs. 2.1 and 2.6 to homogeneous linear ODEs of any order n. This shows that practically all the essential facts carry over without change. Linear independence, now more involved as for n 5 2, causes the Wronskian to become indispensable (whereas for n 5 2 it played a marginal role). Main Content, Important Concepts Superposition principle for the homogeneous ODE (2) General solution, basis, particular solution General solution of (2) with continuous coefficients exists. Existence and uniqueness of solution of initial value problem (2), (5) Linear independence of solutions, Wronskian General solution includes all solutions of (2). Comment on Order of Material In Chap. 2 we first gained practical experience and skill and presented the theory of the homogeneous linear ODE at the end of the discussion, in Sec. 2.6. In this chapter, with all the experience gained on second-order ODEs, it is more logical to present the whole theory at the beginning and the solution methods (for linear ODEs with constant coefficients) afterward. Similarly, the same logic applies to the nonhomogeneous linear ODE, for which Sec. 3.3 contains the theory as well as the solution methods. SOLUTIONS TO PROBLEM SET 3.1, page 111 2. Problems 1–5 should give the student a first impression of the changes occurring in the transition from n 5 2 to general n. 59 im03.qxd 9/21/05 11:04 AM Page 59
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8. Let y 1 5 x 1 1, y 2 5 x 1 2, y 3 5 x. Then y 2 2 2 y 1 1 y 3 5 0 shows linear dependence. 10. Linearly independent 12. Linear dependence, since one of the functions is the zero function 14. cos 2 x 5 cos 2 x 2 sin 2 x ; linearly dependent 16. ( x 2 1) 2 2 ( x 1 1) 2 1 4 x 5 0; linearly dependent 18. Linearly independent 20. Team Project. (a) (1) No. If y 1 ; 0, then (4) holds with any k 1 Þ 0 and the other k j all zero.
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MA441 - CH03 - im03.qxd 9/21/05 11:04 AM Page 59 CHAPTER 3...

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