An introduction to differential equations: With difference eq-s, Fourier ser., and PDEs

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Preface......Page 4
Contents......Page 7
1.1 Introduction and Definitions......Page 11
1.1.1 Applications......Page 13
1.2 Existence and Uniqueness......Page 21
1.3 Variables Separable......Page 27
1.3.1 Applications......Page 30
1.4 First-Order Linear Differential Equations......Page 40
1.4.1 Applications......Page 43
1.5 Exact Differential Equations......Page 51
1.5.1 Application......Page 56
1.6 Homogeneous Equations......Page 61
1.6.1 Application......Page 63
1.7 Equations Reducible to First Order......Page 65
1.7.1 Application......Page 67
Review Exercises......Page 68
2.1 Introduction and Definitions......Page 75
2.1.1 Applications......Page 77
2.2 Linear Independence and Wronskians......Page 83
2.3 Existence and Uniqueness of Solutions......Page 91
2.4 Homogeneous Differential Equations with Constant Coefficients- The Characteristic Equation......Page 97
2.5 Homogeneous Differential Equations with Constant Coefficients- The General Solution......Page 102
2.5.1 Application......Page 107
2.6 Homogeneous Equations with Variable Coefficients- Overview......Page 112
2.7 Euler Differential Equation......Page 113
2.8 Reduction of Order......Page 118
2.8.1 Applications......Page 122
2.9 Solutions of Linear Homogeneous Differential Equations  by the Method of Taylor Series......Page 126
2.10 Nonhomogeneous Differential Equations......Page 130
2.11 The Method of Undetermined Coefficients......Page 134
2.11.1 Applications......Page 139
2.12 Variation of Parameters......Page 145
Review Exercises......Page 151
3.1 Introduction and Basic Theory......Page 157
3.11 Applications......Page 163
3.2 The Method of Elimination......Page 172
3.2.1 Applications......Page 175
3.3 The Matrix Method......Page 180
3.3.1 Nonhomogeneous Systems-Variation of Parameters......Page 191
3.3.2 Applications......Page 193
Review Exercises......Page 197
4.2 The Laplace Transform and Its Properties......Page 200
4.3 The Laplace Transform Applied to Differential Equations  and Systems......Page 209
4.4 The Unit Step Function......Page 216
4.5 The Unit Impulse Function......Page 220
4.6 Applications......Page 224
Review Exercises......Page 231
5.1 Introduction......Page 234
5.2 Review of Power Series......Page 235
5.3 Ordinary Points and Singular Points......Page 239
5.4 Power-Series Solutions about an Ordinary Point......Page 242
5.4.1 Applications......Page 248
5.5 Series Solutions about a Regular Singular Point......Page 256
5.5.1 Applications......Page 268
Review Exercises......Page 276
61 Introduction and Solution of Boundary Value Problems......Page 279
61.1 Applications......Page 283
6.2 Eigenvalues and Eigenfunctions......Page 286
6.2.1 Application......Page 291
Review Exercises......Page 295
7.1 Introduction......Page 296
7.2 Euler Method......Page 298
7.3 Taylor-Series Method......Page 303
7.4 Runge-Kutta Methods......Page 308
7.5 Systems of First-Order Differential Equations......Page 314
7.6 Applications......Page 318
Review Exercises......Page 322
8.2 Existence and Uniqueness Theorems......Page 325
8.3 Solutions and Trajectories of Autonomous Systems......Page 327
8.4 Stability of Critical Points of Autonomous Systems......Page 331
85 Phase Portraits of Autonomous Systems......Page 337
8.6 Applications......Page 347
Review Exercises......Page 356
9.1 Introduction and Definitions......Page 358
9.1.1 Applications......Page 360
9.2 Existence and Uniqueness of Solutions......Page 364
9.3 Linear Independence and the General Solution......Page 368
9.4 Homogeneous Equations with Constant Coefficients......Page 376
9.5.1 Undetermined Coefficients......Page 383
9.5.2 Variation of Parameters......Page 387
9.5.3 Applications......Page 390
Review Exercises......Page 397
10.2 Periodicity and Orthogonality of Sines and Cosines......Page 400
10.3 Fourier Series......Page 404
10.4 Convergence of Fourier Series......Page 411
10.5 Fourier Sine and Fourier Cosine Series......Page 419
Review Exercises......Page 428
11.1 Introduction......Page 431
11.2 Definitions and General Comments......Page 433
11.3 The Principle of Superposition......Page 437
11.4 Separation of Variables......Page 443
11.5 Initial-Boundary Value Problems: An Overview......Page 450
11.6 The Homogeneous One-Dimensional Wave Equation: Separation of Variables......Page 451
11.7 The One-Dimensional Heat Equation......Page 465
11.8 The Potential (Laplace) Equation......Page 471
11.9 Nonhomogeneous Partial Differential Equations: Method I......Page 479
11.10 Nonhomogeneous Partial Differential Equations: Method II......Page 485
Review Exercises......Page 491
Back Cover......Page 495