| Preface | p. ix |
| First-Order Differential Equations and Their Applications | p. 1 |
| Introduction to Ordinary Differential Equations | p. 1 |
| The Definite Integral and the Initial Value Problem | p. 4 |
| The Initial Value Problem and the Indefinite Integral | p. 5 |
| The Initial Value Problem and the Definite Integral | p. 6 |
| Mechanics I: Elementary Motion of a Particle with Gravity Only | p. 8 |
| First-Order Separable Differential Equations | p. 13 |
| Using Definite Integrals for Separable Differential Equations | p. 16 |
| Direction Fields | p. 19 |
| Existence and Uniqueness | p. 25 |
| Euler's Numerical Method (optional) | p. 31 |
| First-Order Linear Differential Equations | p. 37 |
| Form of the General Solution | p. 37 |
| Solutions of Homogeneous First-Order Linear Differential Equations | p. 39 |
| Integrating Factors for First-Order Linear Differential Equations | p. 42 |
| Linear First-Order Differential Equations with Constant Coefficients and Constant Input | p. 48 |
| Homogeneous Linear Differential Equations with Constant Coefficients | p. 48 |
| Constant Coefficient Linear Differential Equations with Constant Input | p. 50 |
| Constant Coefficient Differential Equations with Exponential Input | p. 52 |
| Constant Coefficient Differential Equations with Discontinuous Input | p. 52 |
| Growth and Decay Problems | p. 59 |
| A First Model of Population Growth | p. 59 |
| Radioactive Decay | p. 65 |
| Thermal Cooling | p. 68 |
| Mixture Problems | p. 74 |
| Mixture Problems with a Fixed Volume | p. 74 |
| Mixture Problems with Variable Volumes | p. 77 |
| Electronic Circuits | p. 82 |
| Mechanics II: Including Air Resistance | p. 88 |
| Orthogonal Trajectories (optional) | p. 92 |
| Linear Second- and Higher-Order Differential Equations | p. 96 |
| General Solution of Second-Order Linear Differential Equations | p. 96 |
| Initial Value Problem (for Homogeneous Equations) | p. 100 |
| Reduction of Order | p. 107 |
| Homogeneous Linear Constant Coefficient Differential Equations (Second Order) | p. 112 |
| Homogeneous Linear Constant Coefficient Differential Equations (nth-Order) | p. 122 |
| Mechanical Vibrations I: Formulation and Free Response | p. 124 |
| Formulation of Equations | p. 124 |
| Simple Harmonic Motion (No Damping, [delta] = 0) | p. 128 |
| Free Response with Friction ([delta] > 0) | p. 135 |
| The Method of Undetermined Coefficients | p. 142 |
| Mechanical Vibrations II: Forced Response | p. 159 |
| Friction is Absent ([delta] = 0) | p. 159 |
| Friction is Present ([delta] > 0) (Damped Forced Oscillations) | p. 168 |
| Linear Electric Circuits | p. 174 |
| Euler Equation | p. 179 |
| Variation of Parameters (Second-Order) | p. 185 |
| Variation of Parameters (nth-Order) | p. 193 |
| The Laplace Transform | p. 197 |
| Definition and Basic Properties | p. 197 |
| The Shifting Theorem (Multiplying by an Exponential) | p. 205 |
| Derivative Theorem (Multiplying by t) | p. 210 |
| Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions) | p. 213 |
| Initial Value Problems for Differential Equations | p. 225 |
| Discontinuous Forcing Functions | p. 234 |
| Solution of Differential Equations | p. 239 |
| Periodic Functions | p. 248 |
| Integrals and the Convolution Theorem | p. 253 |
| Derivation of the Convolution Theorem (optional) | p. 256 |
| Impulses and Distributions | p. 260 |
| An Introduction to Linear Systems of Differential Equations and Their Phase Plane | p. 265 |
| Introduction | p. 265 |
| Introduction to Linear Systems of Differential Equations | p. 268 |
| Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix | p. 269 |
| Solving Linear Systems if the Eigenvalues are Real and Unequal | p. 272 |
| Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues | p. 276 |
| Special Systems with Complex Eigenvalues (optional) | p. 279 |
| General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Roots | p. 281 |
| Eigenvalues and Trace and Determinant (optional) | p. 283 |
| The Phase Plane for Linear Systems of Differential Equations | p. 287 |
| Introduction to the Phase Plane for Linear Systems of Differential Equations | p. 287 |
| Phase Plane for Linear Systems of Differential Equations | p. 295 |
| Real Eigenvalues | p. 296 |
| Complex Eigenvalues | p. 304 |
| General Theorems | p. 310 |
| Mostly Nonlinear First-Order Differential Equations | p. 315 |
| First-Order Differential Equations | p. 315 |
| Equilibria and Stability | p. 316 |
| Equilibrium | p. 316 |
| Stability | p. 317 |
| Review of Linearization | p. 318 |
| Linear Stability Analysis | p. 318 |
| One-Dimensional Phase Lines | p. 322 |
| Application to Population Dynamics: The Logistic Equation | p. 327 |
| Nonlinear Systems of Differential Equations in the Plane | p. 332 |
| Introduction | p. 332 |
| Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Plane | p. 335 |
| Linear Stability Analysis and the Phase Plane | p. 336 |
| Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclines | p. 341 |
| Population Models | p. 349 |
| Two Competing Species | p. 350 |
| Predator-Prey Population Models | p. 356 |
| Mechanical Systems | p. 363 |
| Nonlinear Pendulum | p. 363 |
| Linearized Pendulum | p. 364 |
| Conservative Systems and the Energy Integral | p. 364 |
| The Phase Plane and the Potential | p. 367 |
| Answers to Odd-Numbered Exercises | p. 379 |
| Index | p. 429 |
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