UsersGuide - 1 A Guide to Using This Text What follows is a...

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1 A Guide to Using This Text What follows is a suggested schedule of the chapters and sections to be covered (and not covered) in a “typical” introductory course on differential equations, along with some commentary on the material. This discussion is directed towards the instructors teaching such a course. Keep in mind that these are merely suggestions. Each instructor should use their own good judgement and adjust this schedule as appropriate so that their course better suits the backgrounds of their students, the reasons they are taking the course, the time available, and the instructor’s own views on how the course should be taught. Part I: The Basics 1. The Starting Point Cover all of sections 1.1 and 1.2. Section 1.3 should be covered quickly,with the understanding that your students’ understand- ing of and respect for this material will develop as they learn more about differential equations. 2. Integration and Differential Equations Cover sections 2.1 and 2.2 fairly quickly, emphasizing that this is stuff the students should have already seen and be familiar with. Let them know that much of homework is a review of the basic integration methods they will be using extensively for the rest of the course. Do not, however, skip these sections or skip on the homework. Many of your students will probably need the review. It seems that the material in sections 2.3 and 2.4 (on using definite integrals) is rarely part of introductory differential equation courses. It can be skipped. Still, it would not hurt to mention that using definite integrals makes it much easier to numerically solve those directly-integrable differential equations like dy dx = e - x 2 , which require integrating integrals that are not easily integrated. 601
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602 A Guide to Using This Text Part II: First-Order Equations 3. Some Basics about First-Order Equations Sections 3.1 and 3.2 are fundamental and should not be skipped. Section 3.3 (on existence and uniqueness) should only be briefly discussed, and that discussion should probably be limited to theorem 3.1. Most instructors will want to skip the rest of chapter 3. (Still, you might tell the more mathematically inquisitive that the Picard iteration method developed in section 3.4 is pretty cool, and that the discussion is fairly easy to follow since the boring parts have been removed and stuck in the sections 3.5 and 3.6.) 4. Separable First-Order Equations Cover sections 4.1, 4.2, 4.3 and 4.4. You can skip sections 4.5 ( Existence, Uniqueness and False Solutions ), 4.6 ( On the Nature of Solutions to DEs ) and 4.7 ( Using and Graphing Implicit Solutions ). In an idea world, the material in these sections would be recognized as important for understanding “solutions”. But this is not the ideal world and there isn’t enough time to cover everything. Tell your students that they can read it on their own, and that understanding this material will help lead them to enlightenment. Also skip section 4.8. It’s on using definite integrals.
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UsersGuide - 1 A Guide to Using This Text What follows is a...

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