Page 11 - Advanced_Engineering_Mathematics o'neil

Contents

Preface xi

PART 1 Ordinary Differential Equations 1

CHAPTER 1 First-Order Differential Equations 3
1.1 Terminology and Separable Equations 3
1.2 Linear Equations 16
1.3 Exact Equations 21
1.4 Homogeneous, Bernoulli, and Riccati Equations 26
1.4.1 The Homogeneous Differential Equation 26
1.4.2 The Bernoulli Equation 27
1.4.3 The Riccati Equation 28
1.5 Additional Applications 30
1.6 Existence and Uniqueness Questions 40

CHAPTER 2 Linear Second-Order Equations 43
2.1 The Linear Second-Order Equation 43
2.2 The Constant Coefficient Case 50
2.3 The Nonhomogeneous Equation 55
2.3.1 Variation of Parameters 55
2.3.2 Undetermined Coefficients 57
2.3.3 The Principle of Superposition 60
2.4 Spring Motion 61
2.4.1 Unforced Motion 62
2.4.2 Forced Motion 66
2.4.3 Resonance 67
2.4.4 Beats 69
2.4.5 Analogy with an Electrical Circuit 70
2.5 Euler’s Differential Equation 72

CHAPTER 3 The Laplace Transform 77
3.1 Definition and Notation 77
3.2 Solution of Initial Value Problems 81
3.3 Shifting and the Heaviside Function 84

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October 15, 2010 17:43 THM/NEIL Page-v 27410_00_fm_pi-xiv