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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 | |