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| Introduction | |
| Background | |
| Solutions and Initial Value Problems | |
| Direction Fields | |
| The Approximation Method of Euler | |
| First Order Differential Equations | |
| Introduction: Motion of a Falling Body | |
| Separable Equations | |
| Linear Equations | |
| Exact Equations | |
| Special Integrating Factors | |
| Substitutions and Transformations | |
| Mathematical Models and Numerical Methods Involving First Order Equations | |
| Mathematical Modeling | |
| Compartmental Analysis | |
| Heating and Cooling of Buildings | |
| Newtonian Mechanics | |
| Electrical Circuits | |
| Improved Euler's Method | |
| Higher-Order Numerical Methods: Taylor and Runge-Kutta | |
| Linear Second Order Equations | |
| Introduction: The Mass-Spring Oscillator | |
| Homogeneous Linear Equations; the General Solution | |
| Auxiliary Equations with Complex Roots | |
| Nonhomogeneous Equations: the Method of Undetermined Coefficients | |
| The Superposition Principle and Undetermined Coefficients Revisited | |
| Variation of Parameters | |
| Qualitative Considerations for Variable-Coefficient and Nonlinear Equations | |
| A Closer Look at Free Mechanical Vibrations | |
| A Closer Look at Forced Mechanical Vibrations | |
| Introduction to Systems and Phase Plane Analysis | |
| Interconnected Fluid Tanks | |
| Elimination Method for Systems with Constant Coefficients | |
| Solving Systems and Higher-Order Equations Numerically | |
| Introduction to the Phase Plane | |
| Coupled Mass-Spring Systems | |
| Electrical Systems | |
| Dynamical Systems, Poincaré Maps, and Chaos | |
| Theory of Higher-Order Linear Differential Equations | |
| Basic Theory of Linear Differential Equations | |
| Homogeneous Linear Equations with Constant Coefficients | |
| Undetermined Coefficients and the Annihilator Method | |
| Method of Variation of Parameters | |
| Laplace Transforms | |
| Introduction: A Mixing Problem | |
| Definition of the Laplace Transform | |
| Properties of the Laplace Transform | |
| Inverse Laplace Transform | |
| Solving Initial Value Problems | |
| Transforms of Discontinuous and Periodic Functions | |
| Convolution | |
| Impulses and the Dirac Delta Function | |
| Solving Linear Systems with Laplace Transforms | |
| Series Solutions of Differential Equations | |
| Introduction: The Taylor Polynomial Approximation | |
| Power Series and Analytic Functions | |
| Power Series Solutions to Linear Differential Equations | |
| Equations with Analytic Coefficients | |
| Cauchy-Euler (Equidimensional) Equations | |
| Method of Frobenius | |
| Finding a Second Linearly Independent Solution | |
| Special Functions | |
| Matrix Methods for Linear Systems | |
| Introduction | |
| Review 1: Linear Algebraic Equations | |
| Review 2: Matrices and Vectors | |
| Linear Systems in Normal Form | |
| Homogeneous Linear Systems with Constant Coefficients | |
| Complex Eigenvalues | |
| Nonhomogeneous Linear Systems | |
| The Matrix Exponential Function | |
| Partial Differential Equations | |
| Introduction: A Model for Heat Flow | |
| Method of Separation of Variables | |
| Fourier Series | |
| Fourier Cosine and Sine Series | |
| The Heat Equation | |
| The Wave Equation | |
| Laplace's Equation | |
| Eigenvalue Problems and Sturm-Liouville Equations | |
| Introduction: Heat Flow in a Nonuniform Wire | |
| Eigenvalues and Eigenfunctions | |
| Regular Sturm-Liouville Boundary Value Problems | |
| Nonhomogeneous Boundary Value Problems and the Fredholm Alternative | |
| Solution by Eigenfunction Expansion | |
| Green's Functions | |
| Singular Sturm-Liouville Boundary Value Problems | |
| Oscillation and Comparison Theory | |
| Stability of Autonomous Systems | |
| Introduction: Competing Species | |
| Linear Systems in the Plane | |
| Almost Linear Systems | |
| Energy Methods | |
| Lyapunov's Direct Method | |
| Limit Cycles and Periodic Solutions | |
| Stability of Higher-Dimensional Systems | |
| Existence and Uniqueness Theory | |
| Introduction: Successive Approximations | |
| Picard's Existence and Uniqueness Theorem | |
| Existence of Solutions of Linear Equations | |
| Continuous Dependence of Solutions | |
| Appendices. Newton's Method | |
| Simpson's Rule | |
| Cramer's Rule | |
| Method of Least Squares | |
| Runge-Kutta Precedure for n Equations | |
| Answers to Odd-Numbered Problems | |
| Index | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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