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First Order Differential Equations | |
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Differential Equations and Mathematical Models | |
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Integrals as General and Particular Solutions | |
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Slope Fields and Solution Curves | |
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Separable Equations and Applications | |
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Linear First Order Equations | |
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Substitution Methods and Exact Equations | |
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Mathematical Models and Numerical Methods | |
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Population Models | |
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Equilibrium Solutions and Stability | |
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Acceleration-Velocity Models | |
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Numerical Approximation: Euler's Method | |
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A Closer Look at the Euler Method, and Improvements | |
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The Runge-Kutta Method | |
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Linear Equations of Higher Order | |
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Introduction: Second-Order Linear Equations | |
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General Solutions of Linear Equations | |
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Homogeneous Equations with Constant Coefficients | |
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Mechanical Vibrations | |
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Nonhomogeneous Equations and Undetermined Coefficients | |
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Forced Oscillations and Resonance | |
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Electrical Circuits | |
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Endpoint Problems and Eigenvalues | |
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Introduction to Systems of Differential Equations | |
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First-Order Systems and Applications | |
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The Method of Elimination | |
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Numerical Methods for Systems | |
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Linear Systems of Differential Equations | |
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Linear Systems and Matrices | |
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The Eigenvalue Method for Homogeneous Systems | |
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Second Order Systems and Mechanical Applications | |
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Multiple Eigenvalue Solutions | |
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Matrix Exponentials and Linear Systems | |
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Nonhomogenous Linear Systems | |
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Nonlinear Systems and Phenomena | |
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Stability and the Phase Plane | |
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Linear and Almost Linear Systems | |
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Ecological Models: Predators and Competitors | |
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Nonlinear Mechanical Systems | |
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Chaos in Dynamical Systems | |
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Laplace Transform Methods | |
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Laplace Transforms and Inverse Transforms | |
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Transformation of Initial Value Problems | |
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Translation and Partial Fractions | |
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Derivatives, Integrals, and Products of Transforms | |
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Periodic and Piecewise Continuous Forcing Functions | |
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Impulses and Delta Functions | |
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Power Series Methods | |
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Introduction and Review of Power Series | |
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Series Solutions Near Ordinary Points | |
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Regular Singular Points | |
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Method of Frobenius: The Exceptional Cases | |
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Bessel's Equation | |
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Applications of Bessel Functions | |
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Fourier Series Methods | |
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Periodic Functions and Trigonometric Series | |
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General Fourier Series and Convergence | |
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Even-Odd Functions and Termwise Differentiation | |
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Applications of Fourier Series | |
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Heat Conduction and Separation of Variables | |
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Vibrating Strings and the One-Dimensional Wave Equation | |
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Steady-State Temperature and Laplace's Equation | |
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Eigenvalues and Boundary Value Problems | |
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Sturm-Liouville Problems and Eigenfunction Expansions | |
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Applications of Eigenfunction Series | |
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Steady Periodic Solutions and Natural Frequencies | |
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Applications of Bessel Functions | |
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Higher-Dimensional Phenomena | |
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References | |
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Existence and Uniqueness of Solutions | |
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Answers to Selected Problems | |
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Index | |