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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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| Direction Fields and Solutions 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 | |
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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 the Method of 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 Eignvalue Solutions | |
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| Matrix Exponentials and Linear Systems | |
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| Nonhomogeneous 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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| Fourier Sine and Cosine Series | |
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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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| Strum-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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| Cylindrical Coordinate Problems | |
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| Higher-Dimensional Phenomena | |