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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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Appendixes | |
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Gamma Functions | |
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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 | |