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| Preface | |
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| Introduction to Differential Equations | |
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| Differential Equation Models | |
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| The Derivative | |
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| Integration | |
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| First-Order Equations | |
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| Differential Equations and Solutions | |
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| Solutions to Separable Equations | |
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| Models of Motion | |
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| Linear Equations | |
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| Mixing Problems | |
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| Exact Differential Equations | |
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| Existence and Uniqueness of Solutions | |
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| Dependence of Solutions on Initial Conditions | |
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| Autonomous Equations and Stability | |
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| The Daredevil Skydiver | |
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| Modeling and Applications | |
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| Modeling Population Growth | |
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| Models and the Real World | |
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| Personal Finance | |
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| Electrical Circuits | |
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| The Spruce Budworm | |
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| Social Security, Now or Later | |
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| Second-Order Equations | |
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| Definitions and Examples | |
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| Second-Order Equations and Systems | |
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| Linear, Homogeneous Equations with Constant Coefficients | |
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| Harmonic Motion | |
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| Inhomogeneous Equations; the Method of Undetermined Coefficients | |
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| Variation of Parameters | |
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| Forced Harmonic Motion | |
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| Nonlinear Oscillators | |
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| The Laplace Transform | |
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| The Definition of the Laplace Transform | |
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| Basic Properties of the Laplace Transform | |
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| The Inverse Laplace Transform | |
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| Using the Laplace Transform to Solve Differential Equations | |
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| Discontinuous Forcing Terms | |
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| The Delta Function | |
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| Convolutions | |
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| Summary | |
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| Forced Harmonic Oscillators | |
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| Numerical Methods | |
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| Euler's Method | |
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| Runge-Kutta Methods | |
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| Numerical Error Comparisons | |
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| Practical Use of Solvers | |
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| A Cautionary Tale | |
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| Numerical Error Comparison | |
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| Matrix Algebra | |
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| Vectors and Matrices | |
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| The Geometry of Systems of Linear Equations | |
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| Solving Systems of Equations | |
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| Properties of Solution Sets | |
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| Subspaces | |
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| Determinants | |
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| An Introduction to Systems | |
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| Definitions and Examples | |
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| Geometric Interpretation of Solutions | |
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| Qualitative Analysis | |
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| Linear Systems | |
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| Long-Term Behavior of Solutions | |
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| Linear Systems with Constant Coefficients | |
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| Overview of the Technique | |
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| Planar Systems | |
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| Phase Plane Portraits | |
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| Higher Dimensional Systems | |
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| The Exponential of a Matrix | |
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| Qualitative Analysis of Linear Systems | |
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| Higher-Order Linear Equations | |
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| Inhomogeneous Linear Systems | |
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| Phase Plane Portraits | |
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| Oscillations of Linear Molecules | |
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| Nonlinear Systems | |
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| The Linearization of a Nonlinear System | |
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| Long-Term Behavior of Solutions | |
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| Invariant Sets and the Use of Nullclines | |
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| Long-Term Behavior of Solutions to Planar Systems | |
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| Conserved Quantities | |
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| Nonlinear Mechanics | |
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| The Method of Lyapunov | |
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| Predator-Prey Systems | |
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| Human Immune Response to Infectious Disease | |
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| Analysis of Competing Species | |
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| Series Solutions to Differential Equations | |
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| Review of Power Series | |
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| Series Solutions Near Ordinary Points | |
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| Legendre's Equation | |
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| Types of Singular Points--Euler's Equation | |
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| Series Solutions Near Regular Singular Points | |
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| Solutions in the Exceptional Cases | |
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| Bessel's Equation and Bessel Functions | |
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| Fourier Series | |
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| Computation of Fourier Series | |
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| Convergence of Fourier Series | |
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| Fourier Cosine and Sine Series | |
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| The Complex Form of a Fourier Series | |
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| The Discrete Fourier Transform and the FFT | |
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| Partial Differential Equations | |
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| Derivation of the Heat Equation | |
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| Separation of Variables for the Heat Equation | |
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| The Wave Equation | |
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| Laplace's Equation | |
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| Laplace's Equation on a Disk | |
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| Sturm Liouville Problems | |
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| Orthogonality and Generalized Fourier Series | |
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| Temperature in a Ball--Legendre Polynomials | |
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| Time Dependent PDEs in Higher Dimension | |
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| Domains with Circular Symmetry--Bessel Functions | |
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| Answers to Odd-Numbered Problems | |
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| Index | |