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Introduction and First Definitions Modeling via Differential Equations First Order Differential Equations Linear Equations Separable Equations Qualitative Technique: Slope Fields Equilibria and the Phase Line Bifurcations Bernoulli Equations Riccati Equations Homogeneous Equations Exact and Non-Exact Equations Integrating Factor technique Some Applications Radioactive Decay Newton's Law of Cooling Orthogonal Trajectories Population Dynamics Numerical Technique: Euler's Method Existence and Uniqueness of Solutions Picard Iterative Process Second Order Differential Equations Nonlinear Equations Linear Equations Homogeneous Linear Equations Linear Independence and the Wronskian Reduction of Order Homogeneous Equations with Constant Coefficients Non-Homogeneous Linear Equations Method of Undetermined Coefficients Method of Variation of Parameters Euler-Cauchy Equations Series Solutions Introduction Derivatives and Index Shifting First Examples Airy's Equation The Radius of Convergence of Series Solutions Hermite's Equation Higher Order Linear Equations Introduction and Basic Results Homogeneous Linear Equations with Constant Coefficients Non-Homogeneous Linear Equations Method of Undetermined Coefficients Method of Variation of Parameters Laplace Transform Basic Definitions and Results Application to Differential Equations Impulse Functions: Dirac Function Convolution Product Table of Laplace Transforms Systems of Differential Equations Introduction and Motivation Second Order Equations and Systems Euler's Method for Systems Qualitative Analysis Linear Systems Introduction and First Definitions Vector Representations of Solutions of Linear Systems Eigenvalues and Eigenvectors Technique Real Eigenvalues Repeated Eigenvalues Complex Eigenvalues Zero Eigenvalues Qualitative Analysis of Linear Systems Nonlinear Systems Equilibrium Point Analysis: Linearization Technique Fourier Series Fourier Series: Basic Results Fourier Sine and Cosine Series Convergence of Fourier Series More on Convergence of Fourier Series Gibbs Phenomenon Bessel's Inequality and Parseval Formula: The Energy Theorem Operations on Fourier Series Application of Fourier Series to Differential Equations APPENDIX Tables