ch01 - im01.qxd 10:17 AM Page 1 Part A ORDINARY...

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Part A. ORDINARY DIFFERENTIAL EQUATIONS (ODEs) CHAPTER 1 First-Order ODEs Major Changes There is more material on modeling in the text as well as in the problem set. Some additions on population dynamics appear in Sec. 1.5. Electric circuits are shifted to Chap. 2, where second-order ODEs will be available. This avoids repetitions that are unnecessary and practically irrelevant. Team Projects, CAS Projects, and CAS Experiments are included in most problem sets. SECTION 1.1. Basic Concepts. Modeling, page 2 Purpose. To give the students a first impression what an ODE is and what we mean by solving it. Background Material. For the whole chapter we need integration formulas and techniques, which the student should review. General Comments This section should be covered relatively rapidly to get quickly to the actual solution methods in the next sections. Equations (1)–(3) are just examples, not for solution, but the student will see that solutions of (1) and (2) can be found by calculus, and a solution y 5 e x of (3) by inspection. Problem Set 1.1 will help the student with the tasks of Solving y 9 5 ƒ( x ) by calculus Finding particular solutions from given general solutions Setting up an ODE for a given function as solution Gaining a first experience in modeling, by doing one or two problems Gaining a first impression of the importance of ODEs without wasting time on matters that can be done much faster, once systematic methods are available. Comment on “General Solution” and “Singular Solution” Usage of the term “general solution” is not uniform in the literature. Some books use the term to mean a solution that includes all solutions, that is, both the particular and the singular ones. We do not adopt this definition for two reasons. First, it is frequently quite difficult to prove that a formula includes all solutions; hence, this definition of a general solution is rather useless in practice. Second, linear differential equations (satisfying rather general conditions on the coefficients) have no singular solutions (as mentioned in the text), so that for these equations a general solution as defined does include all solutions. For the latter reason, some books use the term “general solution” for linear equations only; but this seems very unfortunate. 1 im01.qxd 9/21/05 10:17 AM Page 1
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SOLUTIONS TO PROBLEM SET 1.1, page 8 2. y 52 e 2 3 x /3 1 c 4. y 5 (sinh 4 x )/4 1 c 6. Second order. 8. First order. 10. y 5 ce 0.5 x , y (2) 5 ce 5 2, c 5 2/ e , y 5 (2/ e ) e 0.5 x 5 0.736 e 0.5 x 12. y 5 ce x 1 x 1 1, y (0) 5 c 1 1 5 3, c 5 2, y 5 2 e x 1 x 1 1 14. y 5 c sec x , y (0) 5 c /cos 0 5 c 5 1 _ 2 p , y 5 1 _ 2 sec x 16. Substitution of y 5 cx 2 c 2 into the ODE gives y 9 2 2 xy 9 1 y 5 c 2 2 xc 1 ( cx 2 c 2 ) 5 0. Similarly, y 5 1 _ 4 x 2 , y 9 5 1 _ 2 x , thus 1 _ 4 x 2 2 x ( 1 _ 2 x ) 1 1 _ 4 x 2 5 0.
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This note was uploaded on 04/27/2008 for the course MA 441 taught by Professor Kaba during the Spring '08 term at Embry-Riddle FL/AZ.

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ch01 - im01.qxd 10:17 AM Page 1 Part A ORDINARY...

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