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Preface | p. ix |
Acknowledgments | p. xv |
Introduction to Differential Equations | p. 1 |
Definitions and Terminology | p. 2 |
Initial-Value Problems | p. 15 |
Differential Equations as Mathematical Models | p. 22 |
Chapter 1 in Review | p. 37 |
First-Order Differential Equations | p. 39 |
Solution Curves Without the Solution | p. 40 |
Separable Variables | p. 51 |
Linear Equations | p. 60 |
Exact Equations | p. 72 |
Solutions by Substitutions | p. 80 |
A Numerical Solution | p. 86 |
Chapter 2 in Review | p. 92 |
Modeling with First-Order Differential Equations | p. 95 |
Linear Equations | p. 96 |
Nonlinear Equations | p. 109 |
Systems of Linear and Nonlinear Differential Equations | p. 121 |
Chapter 3 in Review | p. 130 |
Project Module: Harvesting of Renewable Natural Resources | p. 133 |
Higher-Order Differential Equations | p. 138 |
Preliminary Theory: Linear Equations | p. 139 |
Initial-Value and Boundary-Value Problems | p. 139 |
Homogeneous Equations | p. 142 |
Nonhomogeneous Equations | p. 148 |
Reduction of Order | p. 154 |
Homogeneous Linear Equations with Constant Coefficients | p. 158 |
Undetermined Coefficients--Superposition Approach | p. 167 |
Undetermined Coefficients--Annihilator Approach | p. 178 |
Variation of Parameters | p. 188 |
Cauchy-Euler Equation | p. 193 |
Solving Systems of Linear Equations by Elimination | p. 201 |
Nonlinear Equations | p. 207 |
Chapter 4 in Review | p. 212 |
Modeling with Higher-Order Differential Equations | p. 215 |
Linear Equations: Initial-Value Problems | p. 216 |
Spring/Mass Systems: Free Undamped Motion | p. 216 |
Spring/Mass Systems: Free Damped Motion | p. 220 |
Spring/Mass Systems: Driven Motion | p. 224 |
Series Circuit Analogue | p. 227 |
Linear Equations: Boundary-Value Problems | p. 237 |
Nonlinear Equations | p. 247 |
Chapter 5 in Review | p. 259 |
Project Module: The Collapse of the Tacoma Narrows Suspension Bridge | p. 263 |
Series Solutions of Linear Equations | p. 267 |
Solutions About Ordinary Points | p. 268 |
Review of Power Series | p. 268 |
Power Series Solutions | p. 271 |
Solutions About Singular Points | p. 280 |
Two Special Equations | p. 292 |
Chapter 6 in Review | p. 304 |
The Laplace Transform | p. 306 |
Definition of the Laplace Transform | p. 307 |
Inverse Transform and Transforms of Derivatives | p. 314 |
Translation Theorems | p. 324 |
Translation on the s-Axis | p. 324 |
Translation on the t-Axis | p. 328 |
Additional Operational Properties | p. 338 |
Dirac Delta Function | p. 351 |
Systems of Linear Equations | p. 354 |
Chapter 7 in Review | p. 361 |
Systems of Linear First-Order Differential Equations | p. 364 |
Preliminary Theory | p. 365 |
Homogeneous Linear Systems with Constant Coefficients | p. 375 |
Distinct Real Eigenvalues | p. 376 |
Repeated Eigenvalues | p. 380 |
Complex Eigenvalues | p. 384 |
Variation of Parameters | p. 393 |
Matrix Exponential | p. 399 |
Chapter 8 in Review | p. 404 |
Project Module: Earthquake Shaking of Multistory Buildings | p. 406 |
Numerical Solutions of Ordinary Differential Equations | p. 410 |
Euler Methods and Error Analysis | p. 411 |
Runge-Kutta Methods | p. 417 |
Multistep Methods | p. 424 |
Higher-Order Equations and Systems | p. 427 |
Second-Order Boundary-Value Problems | p. 433 |
Chapter 9 in Review | p. 438 |
Appendixes | p. 1 |
Gamma Functions | p. 1 |
Introduction to Matrices | p. 3 |
Laplace Transforms | p. 25 |
Selected Answers for Odd-Numbered Problems | p. 1 |
Index | p. 1 |
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