TOC - Contents Authors Foreword . . . . . . . . . . . . . ....

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Contents Author’s Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii I The Basics 1 1 The Starting Point: Basic Concepts and Terminology 3 1.1 Differential Equations: Basic Definitions and Classifications . . . . . . . . . . 3 Solutions: The Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . 5 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Why Care About Differential Equations? Some Illustrative Examples . . . . . 8 The Situation to Be Modeled: . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Simplest Falling Object Model . . . . . . . . . . . . . . . . . . . . . . 10 A Better Falling Object Model . . . . . . . . . . . . . . . . . . . . . . . . . 11 Summary of Our Models and the Related Initial Value Problems . . . . . . . 14 1.3 More on Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Intervals of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Solutions Over Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Describing Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . 17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Integration and Differential Equations 23 2.1 Directly-Integrable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 On Using Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 On Using Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Advantages of Using Definite Integrals . . . . . . . . . . . . . . . . . . . . 28 Important “Named" Definite Integrals with Variable Limits . . . . . . . . . . 29 2.4 Integrals of Piecewise-Defined Functions . . . . . . . . . . . . . . . . . . . . 31 Computing the Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Continuity of the Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 II First-Order Equations 41 3 Some Basics about First-Order Equations 43 3.1 Algebraically Solving for the Derivative . . . . . . . . . . . . . . . . . . . . . 43 iii
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iv 3.2 Constant (or Equilibrium) Solutions . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 On the Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . 48 3.4 Confirming the Existence of Solutions (Core Ideas) . . . . . . . . . . . . . . . 50 Converting to an Integral Equation . . . . . . . . . . . . . . . . . . . . . . . 50 Generating a Sequence of “Approximate Solutions” . . . . . . . . . . . . . . 51 (Naively) Taking the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Details in the Proofs of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 53 Confirming the Existence of Solutions . . . . . . . . . . . . . . . . . . . . . 53
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TOC - Contents Authors Foreword . . . . . . . . . . . . . ....

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