# Ordinary And Partial Differential Equations For The Beginner

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ISBN-13: 9789814723992 World Scientific Publishing Company, Incorporated 05/24/2016 Reprint 256 6.00(w) x 8.90(h) x 0.60(d)

Preface xiii

1 Ordinary Differential Equations 1

1.1 Basic concepts and terminology 1

1.2 Problems 4

1.3 Auxiliary results from functional analysis 5

1.4 Problems 7

1.5 Approximate solutions. Peano's theorem 8

1.6 Problems 13

1.7 Existence and uniqueness 13

1.8 Problems 18

1.9 Parametric differential equations 19

1.10 Problems 26

1.11 Characteristic function 26

1.12 Problems 28

2 Elementary Solution Methods 29

2.1 Separable differential equations 29

2.2 Problems 31

2.3 Differential equations of homogeneous degree 32

2.4 Problems 35

2.5 First order linear differential equations 36

2.6 Problems 38

2.7 Bernoulli equations 39

2.8 Problems 39

2.9 Riccati equations 40

2.10 Problems 40

2.11 Exact differential equations 41

2.12 Problems 45

2.13 Incomplete differential equations 46

2.14 Problems 50

2.15 Implicit differential equations 50

2.16 Problems 52

2.17 Lagrange and Clairut equations 52

2.18 Problems 54

3 Linear Differential Equations 55

3.1 Integrals of linear differential equations 55

3.2 Problems 59

3.3 Linear differential equations with constant coefficients 59

3.4 Problems 61

3.5 Computation of the exponential matrix 63

3.6 Problems 66

4 Functional Dependence, Independence 69

4.1 Functional independence 69

4.2 Functional expressibility 72

4.3 First integrals 74

4.4 Problems 76

5 Higher Order Differential Equations 77

5.1 A reduction principle 77

5.2 Problems 79

5.3 Intermediate integrals 79

5.4 Problems 83

5.5 Higher order linear differential equations 83

5.6 Problems 85

5.7 Linear differential equations with constant coefficients 86

5.8 Problems 87

5.9 Decreasing the order of linear homogeneous equations 88

5.10 Problems 91

5.11 Euler differential equations 91

5.12 Problems 92

5.13 Exponential polynomials 92

5.14 Problems 95

5.15 Boundary value problems 95

5.16 Problems 101

5.17 Power series solutions 101

5.18 Problems 106

5.19 The Laplace transform 106

5.20 Problems 109

5.21 The Fourier transform of exponential polynomials 110

5.22 Problems 118

6 First Order Partial Differential Equations 119

6.1 Homogeneous linear partial differential equations 119

6.2 Problems 123

6.3 Quasilinear partial differential equations 124

6.4 Problems 127

7 Theory of Characteristics 129

7.1 First order partial differential equations 129

7.2 Problems 131

7.3 Cauchy problem for first order equations 132

7.4 Problems 139

7.5 Special Cauchy problem for first order partial differential equation 139

7.6 Problems 144

7.7 Complete integral 144

7.8 Problems 150

8 Higher Order Partial Differential Equations 151

8.1 Special Cauchy problems for higher order partial differential equations 151

8.2 Theorems of Kovalevskaya and Holmgren 153

8.3 Linear partial differential operators 154

8.4 Problems 158

8.5 Exponential polynomial solutions of partial differential equations 158

8.6 Problems 163

9 Second Order Quasilinear Partial Differential Equations 165

9.1 Second order partial differential equations with linear principal part 165

9.2 Problems 166

9.3 Linear transformation, normal form 167

9.4 Problems 168

9.5 Reduced normal form 169

9.6 Problems 172

9.7 Normal form in two variables 173

9.8 Hyperbolic equations 174

9.9 Problems 176

9.10 Parabolic equations 177

9.11 Problems 178

9.12 Elliptic equations 179

9.13 Problems 180

10 Special Problems in Two Variables 185

10.1 Goursat problem for hyperbolic equations 185

10.2 Problems 188

10.3 Cauchy problem for hyperbolic equations 189

10.4 Problems 192

10.5 Mixed problem for the wave equation 192

10.6 Problems 195

10.7 Fourier method 195

10.8 Problems 197

11 Table of Laplace Transforms 199

12 Answers to Selected Problems 203

1.2 Basic concepts and terminology 203

1.10 Parametric differential equations 203

1.12 Characteristic function 203

2.2 Separable differential equations 204

2.4 Differential equations of homogeneous degree 204

2.6 First order linear differential equations 205

2.8 Bernoulli equations 205

2.10 Riccati equations 206

2.12 Exact differential equations 206

2.14 Incomplete differential equations 207

2.16 Implicit differential equations 208

2.15 Lagrange and Clairut equations 210

3.2 Integrals of linear differential equations 211

3.4 Linear differential equations with constant coefficients 211

3.6 Computation of the exponential matrix 213

4.4 First integrals 214

5.4 Intermediate integrals 214

5.6 Higher order linear differential equations 215

5.8 Linear differential equations with constant coefficients 215

5.10 Decreasing the order of linear homogeneous equations 216

5.12 Euler differential equations 216

5.14 Exponential polynomials 216

5.16 Boundary value problems 216

5.20 Power series solutions 217

5.22 The Laplace transform 218

5.24 The Fourier transform of exponential polynomials 218

6.2 Homogeneous linear partial differential equations 219

6.4 Quasilinear partial differential equations 219

7.2 First order partial differential equations 220

7.4 Cauchy problem for first, order equations 221

7.6 Special Cauchy problem for first order partial differential equation 221

7.8 Complete integral 221

8.6 Exponential polynomial solutions of partial differential equations 222

9.2 Second order partial differential equations with linear principal part 222

9.4 Linear transformation, normal form 223

9.6 Reduced normal form 224

9.9 Hyperbolic equations 225

9.11 Parabolic equations 225

9.13 Elliptic equations 225

10.2 Goursat problem for hyperbolic equations 229

10.4 Cauchy problem for hyperbolic equations 229

10.6 Mixed problem for the wave equation 229

10.8 Fourier method 229

Bibliography 231

Index 233

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