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| Preface | |
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| Acknowledgments | |
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| Introduction to Differential Equations | |
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| Definitions and Terminology | |
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| Initial-Value Problems | |
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| Differential Equations as Mathematical Models | |
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| Chapter 1 in Review | |
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| First-Order Differential Equations | |
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| Solution Curves Without the Solution | |
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| Separable Variables | |
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| Linear Equations | |
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| Exact Equations | |
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| Solutions by Substitutions | |
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| A Numerical Solution | |
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| Chapter 2 in Review | |
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| Modeling with First-Order Differential Equations | |
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| Linear Equations | |
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| Nonlinear Equations | |
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| Systems of Linear and Nonlinear Differential Equations | |
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| Chapter 3 in Review | |
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| Project Module: Harvesting of Renewable Natural Resources | |
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| Higher-Order Differential Equations | |
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| Preliminary Theory: Linear Equations | |
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| Initial-Value and Boundary-Value Problems | |
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| Homogeneous Equations | |
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| Nonhomogeneous Equations | |
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| Reduction of Order | |
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| Homogeneous Linear Equations with Constant Coefficients | |
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| Undetermined Coefficients--Superposition Approach | |
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| Undetermined Coefficients--Annihilator Approach | |
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| Variation of Parameters | |
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| Cauchy-Euler Equation | |
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| Solving Systems of Linear Equations by Elimination | |
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| Nonlinear Equations | |
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| Chapter 4 in Review | |
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| Modeling with Higher-Order Differential Equations | |
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| Linear Equations: Initial-Value Problems | |
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| Spring/Mass Systems: Free Undamped Motion | |
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| Spring/Mass Systems: Free Damped Motion | |
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| Spring/Mass Systems: Driven Motion | |
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| Series Circuit Analogue | |
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| Linear Equations: Boundary-Value Problems | |
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| Nonlinear Equations | |
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| Chapter 5 in Review | |
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| Project Module: The Collapse of the Tacoma Narrows Suspension Bridge | |
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| Series Solutions of Linear Equations | |
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| Solutions About Ordinary Points | |
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| Review of Power Series | |
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| Power Series Solutions | |
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| Solutions About Singular Points | |
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| Two Special Equations | |
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| Chapter 6 in Review | |
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| The Laplace Transform | |
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| Definition of the Laplace Transform | |
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| Inverse Transform and Transforms of Derivatives | |
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| Translation Theorems | |
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| Translation on the s-Axis | |
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| Translation on the t-Axis | |
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| Additional Operational Properties | |
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| Dirac Delta Function | |
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| Systems of Linear Equations | |
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| Chapter 7 in Review | |
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| Systems of Linear First-Order Differential Equations | |
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| Preliminary Theory | |
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| Homogeneous Linear Systems with Constant Coefficients | |
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| Distinct Real Eigenvalues | |
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| Repeated Eigenvalues | |
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| Complex Eigenvalues | |
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| Variation of Parameters | |
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| Matrix Exponential | |
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| Chapter 8 in Review | |
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| Project Module: Earthquake Shaking of Multistory Buildings | |
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| Numerical Solutions of Ordinary Differential Equations | |
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| Euler Methods and Error Analysis | |
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| Runge-Kutta Methods | |
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| Multistep Methods | |
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| Higher-Order Equations and Systems | |
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| Second-Order Boundary-Value Problems | |
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| Chapter 9 in Review | |
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| Plane Autonomous Systems and Stability | |
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| Autonomous Systems, Critical Points, and Periodic Solutions | |
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| Stability of Linear Systems | |
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| Linearization and Local Stability | |
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| Modeling Using Autonomous Systems | |
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| Chapter 10 in Review | |
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| Orthogonal Functions and Fourier Series | |
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| Orthogonal Functions | |
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| Fourier Series | |
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| Fourier Cosine and Sine Series | |
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| Sturm-Liouville Problem | |
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| Bessel and Legendre Series | |
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| Fourier-Bessel Series | |
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| Fourier-Legendre Series | |
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| Chapter 11 in Review | |
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| Partial Differential Equations and Boundary-Value Problems in Rectangular Coordinates | |
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| Separable Partial Differential Equations | |
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| Classical Equations and Boundary-Value Problems | |
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| Heat Equation | |
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| Wave Equation | |
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| Laplace's Equation | |
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| Nonhomogeneous Equations and Boundary Conditions | |
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| Orthogonal Series Expansions | |
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| Boundary-Value Problems Involving Fourier Series in Two Variables | |
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| Chapter 12 in Review | |
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| Boundary-Value Problems in other Coordinate Systems | |
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| Problems Involving Laplace's Equation in Polar Coordinates | |
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| Problems in Polar and Cylindrical Coordinates: Bessel Functions | |
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| Problems in Spherical Coordinates: Legendre Polynomials | |
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| Chapter 13 in Review | |
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| Integral Transform Method | |
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| Error Function | |
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| Applications of the Laplace Transform | |
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| Fourier Integral | |
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| Fourier Transforms | |
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| Chapter 14 in Review | |
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| Numerical Solutions of Partial Differential Equations | |
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| Elliptic Equations | |
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| Parabolic Equations | |
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| Hyperbolic Equations | |
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| Chapter 15 in Review | |
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| Appendixes | |
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| Gamma Function | |
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| Introduction to Matrices | |
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| Laplace Transforms | |
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| Selected Answers for Odd-Numbered Problems | |
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| Index | |