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Introduction To Differential Equations | |
Basic Definitions and Terminology | |
Some Mathematical Models | |
Review | |
Exercises | |
First-Order Differential Equations | |
Preliminary Theory | |
Separable Variables | |
Homogeneous Equations | |
Exact Equations | |
Linear Equations | |
Equations of Bernoulli, Ricatti, and Clairaut | |
Substitutions | |
Picard's Method | |
Review | |
Exercises | |
Applications Of First-Order Differential Equations | |
Orthogonal Trajectories | |
Applications of Linear Equations | |
Applications of Nonlinear Equations | |
Review | |
Exercises | |
Essay: Population Dynamics | |
Linear Differential Equations Of Higher-Order | |
Preliminary Theory | |
Constructing a Second Solution from a Know Solution | |
Homogeneous Linear Equations with Constant Coefficients | |
Undetermined Coefficients: Superposition Approach | |
Differential Operators | |
Undetermined Coefficients: Annihilator Approach | |
Variation of Parameters | |
Review | |
Exercises | |
Essay: Chaos | |
Applications Of Second-Order Differential Equations: Vibrational Models | |
Simple Harmonic Motion | |
Damped Motion | |
Forced Motion | |
Electric Circuits and Other Analogous Systems | |
Review | |
Exercises | |
Essay: Tacoma Narrows Suspension Bridge Collapse | |
Differential Equations With Variable Coefficients | |
Cauchy-Euler Equation | |
Review of Power Series; Power Series Solutions | |
Solutions About Ordinary Points | |
Solutions About Singular Points | |
Two Special Equations | |
Review | |
Exercises | |
Laplace Transform | |
Laplace Transform | |
Inverse Transform | |
Translation Theorems and Derivatives of a Transform | |
Transforms of Derivatives, Integrals, and Periodic Functions | |
Applications | |
Dirac Delta Function | |
Review | |
Exercises | |
Systems Of Linear Differential Equations | |
Operator Method | |
Laplace Transform Method | |
Systems of Linear First-Order Equations | |
Introduction to Matrices | |
Matrices and Systems of Linear First-Order Equations | |
Homogeneous Linear Systems | |
Undetermined Coefficients | |
Variation of Parameters | |
Matrix Exponential | |
Review | |
Exercises | |
Numerical Methods For Ordinary Differential Equations | |
Direction Fields | |
The Euler Methods | |
The Three-Term Taylor Method | |
The Runge-Kutta Methods | |
Multistep Methods | |
Errors and Stability | |
Higher-Order Equations and Systems | |
Second-Order Boundary-Value Problems | |
Review | |
Exercises | |
Essay: Nerve Impulse Models | |
Gamma Function | |
Laplace Transforms | |
Review Of Determinants | |
Complex Numbers | |
Answers To Odd-Numbered Problems | |
Table of Contents provided by Publisher. All Rights Reserved. |
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1. INTRODUCTION TO DIFFERENTIAL EQUATIONS Basic Definitions and Terminology / Some Mathematical Models / Review / Exercises 2. FIRST-ORDER DIFFERENTIAL EQUATIONS Preliminary Theory / Separable Variables / Homogeneous Equations / Exact Equations / Linear Equations / Equations of Bernoulli, Ricatti, and Clairaut / Substitutions / Picard's Method / Review / Exercises 3. APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS Orthogonal Trajectories / Applications of Linear Equations / Applications of Nonlinear Equations / Review / Exercises / Essay: Population Dynamics 4. LINEAR DIFFERENTIAL EQUATIONS OF HIGHER-ORDER Preliminary Theory / Constructing a Second Solution from a Know Solution / Homogeneous Linear Equations with Constant Coefficients / Undetermined Coefficients: Superposition Approach / Differential Operators / Undetermined Coefficients: Annihilator Approach / Variation of Parameters / Review / Exercises / Essay: Chaos 5. APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS: VIBRATIONAL MODELS Simple Harmonic Motion / Damped Motion / Forced Motion / Electric Circuits and Other Analogous Systems / Review / Exercises / Essay: Tacoma Narrows Suspension Bridge Collapse 6. DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS Cauchy-Euler Equation / Review of Power Series; Power Series Solutions / Solutions About Ordinary Points / Solutions About Singular Points / Two Special Equations / Review / Exercises 7. LAPLACE TRANSFORM Laplace Transform / Inverse Transform / Translation Theorems and Derivatives of a Transform / Transforms of Derivatives, Integrals, and Periodic Functions / Applications / Dirac Delta Function / Review / Exercises 8. SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS Operator Method / Laplace Transform Method / Systems of Linear First-Order Equations / Introduction to Matrices / Matrices and Systems of Linear First-Order Equations / Homogeneous Linear Systems / Undetermined Coefficients / Variation of Parameters / Matrix Exponential / Review / Exercises 9. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS Direction Fields / The Euler Methods / The Three-Term Taylor Method / The Runge-Kutta Methods / Multistep Methods / Errors and Stability / Higher-Order Equations and Systems / Second-Order Boundary-Value Problems / Review / Exercises / Essay: Nerve Impulse Models / APPENDIX I: GAMMA FUNCTION / APPENDIX II: LAPLACE TRANSFORMS / APPENDIX III: REVIEW OF DETERMINANTS / APPENDIX IV: COMPLEX NUMBERS / ANSWERS TO ODD-NUMBERED PROBLEMS
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