| Preface | |
| Ch. 1 | Basic Concepts | 1 |
| 1.1 | Simple Differential Equations and Explicit Solutions | 1 |
| 1.2 | Graphical Solutions Using Calculus | 7 |
| 1.3 | Slope Fields and Isoclines | 11 |
| 1.4 | Functions and Power Series Expansions | 22 |
| Ch. 2 | Autonomous Differential Equations | 27 |
| 2.1 | Autonomous Equations | 28 |
| 2.2 | Simple Models | 36 |
| 2.3 | The Logistic Equation | 42 |
| 2.4 | Existence and Uniqueness of Solutions, and Words of Caution | 53 |
| 2.5 | Qualitative Behavior of Solutions Using Phase Lines | 65 |
| 2.6 | Bifurcation Diagrams | 75 |
| Ch. 3 | First Order Differential Equations - Qualitative and Quantitative Aspects | 81 |
| 3.1 | Graphical Solutions Using Calculus | 81 |
| 3.2 | Symmetry of Slope Fields | 94 |
| 3.3 | Numerical Solutions and Chaos | 97 |
| 3.4 | Comparing Solutions of Differential Equations | 112 |
| 3.5 | Finding Power Series Solutions | 115 |
| Ch. 4 | Models and Applications Leading to New Techniques | 119 |
| 4.1 | Solving Separable Differential Equations | 119 |
| 4.2 | Solving Differential Equations with Homogeneous Coefficients | 136 |
| 4.3 | Models: Deriving Differential Equations From Data | 145 |
| 4.4 | Models: Objects in Motion | 158 |
| 4.5 | Application: Orthogonal Trajectories | 167 |
| 4.6 | Piecing Together Differential Equations | 175 |
| Ch. 5 | First Order Linear Differential Equations and Models | 183 |
| 5.1 | Solving Linear Differential Equations | 183 |
| 5.2 | Models That Use Linear Equations | 196 |
| 5.3 | Models That Use Bernoulli's Equation | 211 |
| Ch. 6 | Interplay Between First Order Systems and Second Order Equations | 223 |
| 6.1 | Simple Models | 223 |
| 6.2 | How First Order Systems and Second Order Equations Are Related | 231 |
| 6.3 | Second Order Linear Differential Equations with Constant Coefficients | 241 |
| 6.4 | Modeling Physical Situations | 250 |
| 6.5 | Interpreting the Phase Plane | 259 |
| 6.6 | How Explicit Solutions Are Related to Orbits | 266 |
| 6.7 | The Motion of a Nonlinear Pendulum | 272 |
| Ch. 7 | Second Order Linear Differential Equations with Forcing Functions | 279 |
| 7.1 | The General Solution | 279 |
| 7.2 | Finding Solutions by the Method of Undetermined Coefficients | 283 |
| 7.3 | Application and Models | 294 |
| Ch. 8 | Second Order Linear Differential Equations - Qualitative and Quantitative Aspects | 315 |
| 8.1 | Qualitative Behavior of Solutions | 315 |
| 8.2 | Finding Solutions by Reduction of Order | 323 |
| 8.3 | Finding Solutions by Variation of Parameters | 330 |
| 8.4 | The Importance of Linear Independence and Dependence | 336 |
| 8.5 | Solving Cauchy-Euler Equations | 343 |
| 8.6 | Boundary Value Problems and the Shooting Method | 350 |
| 8.7 | Solving Higher Order Homogeneous Differential Equations | 355 |
| 8.8 | Solving Higher Order Nonhomogeneous Differential Equations | 368 |
| Ch. 9 | Linear Autonomous Systems | 379 |
| 9.1 | Solving Linear Autonomous Systems | 379 |
| 9.2 | Classification of Solutions via Stability | 388 |
| 9.3 | When Do Straight-Line Orbits Exist? | 403 |
| 9.4 | Qualitative Behavior Using Nullclines | 408 |
| 9.5 | Matrix Formulation of Solutions | 413 |
| 9.6 | Compartmental Models | 430 |
| Ch. 10 | Nonlinear Autonomous Systems | 315 |
| 10.1 | Introduction to Nonlinear Autonomous Systems | 437 |
| 10.2 | Qualitative Behavior Using Nullcline Analysis | 448 |
| 10.3 | Qualitative Behavior Using Linearization | 456 |
| 10.4 | Models Involving Nonlinear Autonomous Equations | 467 |
| 10.5 | Bungee Jumping | 485 |
| 10.6 | Linear Versus Nonlinear Differential Equations | 492 |
| 10.7 | Autonomous Versus Nonautonomous Differential Equations | 498 |
| Ch. 11 | Using Laplace Transforms | 505 |
| 11.1 | Motivation | 505 |
| 11.2 | Constructing New Laplace Transforms from Old | 516 |
| 11.3 | The Inverse Laplace Transform and the Convolution Theorem | 523 |
| 11.4 | Functions That Jump | 531 |
| 11.5 | Models Involving First Order Linear Differential Equations | 545 |
| 11.6 | Models Involving Higher Order Linear Differential Equations | 553 |
| 11.7 | Applications to Systems of Linear Differential Equations | 560 |
| 11.8 | When Do Laplace Transforms Exist? | 566 |
| Ch. 12 | Using Power Series | 575 |
| 12.1 | Solutions Using Taylor Series | 576 |
| 12.2 | Solutions Using Power Series | 581 |
| 12.3 | What To Do When Power Series Fail | 599 |
| 12.4 | Solutions Using the Method of Frobenius | 609 |
| Appendices | 631 |
| A.1 | Background Material | 631 |
| A.2 | Partial Fractions | 637 |
| A.3 | Infinite Series, Power Series, and Taylor Series | 640 |
| A.4 | Complex Numbers | 642 |
| A.5 | Elementary Matrix Operations | 645 |
| A.6 | Least Squares Approximation | 651 |
| A.7 | Proofs of the Oscillation Theorems | 654 |
| Answers | 659 |
| Index | 672 |