Table of Contents
Preface v
Chapter 1 First-Order Differential Equations 1
1.1 Definition of Differential Equations 1
1.2 Mathematical Models 9
1.2.1 Newton's Law of Cooling 5
1.2.2 Newton's Law of Motion 11
1.2.3 Torricelli's Law for Draining 13
1.2.4 Population Models 17
1.2.5 A Swimmer's Problem 18
1.2.6 Slope Fields & Solution Curves 21
1.3 Separation of Variables 26
1.4 Linear First-Order DEs 32
1.5 Substitution Methods 40
1.5.1 Polynomial Substitution 40
1.5.2 Homogeneous DEs 42
1.5.3 Bernoulli DEs 44
1.6 The Exact DEs 51
1.7 Riccati DEs 72
Chapter 2 Mathematical Models 76
2.1 Population Model 76
2.1.1 General Population Equation 76
2.1.2 The Logistic Equation 78
2.1.3 Doomsday vs. Extinction 81
2.2 Acceleration-Velocity Model 87
2.2.1 Velocity and Acceleration Models 87
2.2.2 Air Resistance Model 88
2.2.3 Gravitational Acceleration 93
2.3 An Example in Finance 101
Chapter 3 Linear Des of Higher Order 107
3.1 Classification of Des 107
3.2 Linear Independence 112
3.3 Constant Coefficient Homogeneous DEs 121
3.4 Cauchy-Euler DEs 135
3.5 inhomogeneous higher order des 139
3.6 Variation of Parameters 151
Chapter 4 Systems of Linear Des 162
4.1 Basics of systems 162
4.2 First-Order Systems and Applications 165
4.3 Substitution Method 173
4.4 Operator Method 179
4.5 Eigen-Analysis Method 184
Chapter 5 Laplace Transforms 193
5.1 Laplace Transforms 193
5.2 Properties of Laplace Transforms 195
5.2.1 Laplace Transforms for Polynomials 196
5.2.2 The Translator Property 199
5.2.3 Shifting Property 203
5.2.4 The t-multiplication property 206
5.2.5 Periodic Functions 209
5.2.6 Differentiation and Integration Property 210
5.3 Inverse Laplace Transforms 215
5.4 The Convolution of Two Functions 220
5.5 Application of Laplace Transforms 224
Appendix A Solutions to Selected Problems 238
Chapter 1 First-Order DEs 238
1.1 Definition of DEs 238
1.2 Mathematical Models 244
1.3 Separation of Variables 254
1.4 Linear First-Order DEs 262
1.5 Substitution Methods 273
1.6 The Exact DEs 296
1.7 Riccati DEs 308
Chapter 2 Mathematical Models 316
2.1 Population Model 316
2.2 Acceleration-Velocity Model 326
2.3 An example in Finance 358
Chapter 3 Linear DEs of Higher Order 368
3.1 Classification of DEs 368
3.2 Linear Independence 370
3.3 Constant Coefficient Homogeneous DEs 378
3.4 Cauchy-Euler DEs 391
3.5 Inhomogeneous Higher Order DEs 396
3.6 Variation of Parameters 415
Chapter 4 Systems of Linear DEs 426
4.2 First-Order Systems and Applications 426
4.3 Substitution Method 431
4.4 Operator Method 439
4.5 Eigen-Analysis Method 447
Chapter 5 Laplace Transforms 455
5.2 Properties of Laplace Transforms 455
5.3 Inverse Laplace Transforms 463
5.4 The Convolution of Two Functions 467
5.5 Application of Laplace Transforms 470
Appendix B Laplace Transforms 505
Selected Laplace Transforms 505
Selected Properties of Laplace Transforms 506
Appendix C Derivatives & Integrals 509
Appendix D Abbreviations 511
Appendix E Teaching Plans 513
References 515
Index 517
