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| Preface | |
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| Modeling with Differential Equations | |
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| Terminology | |
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| Differential Equations Describing Populations | |
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| Remarks on Modeling with Differential Equations | |
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| Newton's Law of Cooling | |
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| Loaded Horizontal Beams | |
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| Some Special First Order Ordinary Differential Equations | |
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| Separable Differential Equations | |
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| Linear First Order Differential Equations | |
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| Bernoulli Equations | |
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| Homogeneous Equations | |
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| Exact Differential Equations | |
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| Projects | |
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| How to Review and Remember | |
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| Review of First Order Differential Equations | |
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| Before Module 3 Oscillating Systems and Hanging Cables | |
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| Spring-Mass-Systems | |
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| LRC Circuits | |
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| The Simple Pendulum | |
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| Suspended Cables | |
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| Projects | |
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| Linear Differential Equations with Constant Coefficients | |
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| Homogeneous Linear Differential Equations with Constant Coefficients | |
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| Solving Initial and Boundary Value Problems | |
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| Designing Oscillating Systems | |
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| The Method of Undetermined Coefficients | |
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| Variation of Parameters | |
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| Cauchy-Euler Equations | |
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| Some Results on Boundary Value Problems | |
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| Projects | |
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| Qualitative and Numerical Analysis of Differential Equations | |
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| Direction Fields and Autonomous Equations | |
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| From Visualization to Algorithm: Euler's Method | |
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| Runge-Kutta Methods | |
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| Finite Difference Methods for Second Order Boundary Value Problems | |
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| Linear Differential Equations-Theory | |
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| Existence and Uniqueness of Solutions | |
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| Linear Independence for Vectors | |
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| Matrices and Determinants | |
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| Linear Independence for Functions | |
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| The General Solution of Homogeneous Equations | |
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| Before Module 6 Coupled Electrical and Mechanical Systems | |
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| Multi-Loop Circuits and Kirchhoff's Laws | |
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| Coupled Spring-Mass-Systems | |
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| Laplace Transforms | |
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| Introducing the Laplace Transform | |
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| Solving Differential Equations with Laplace Transforms | |
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| Systems of Linear Differential Equations | |
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| Expanding the Transform Table | |
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| Discontinuous Forcing Terms | |
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| Complicated Forcing Functions and Convolutions | |
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| Projects | |
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| Before Module 7 Vibration and Heat | |
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| Vibrating Strings | |
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| The Heat Equation | |
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| The Schrodinger Equation | |
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| Introduction to Partial Differential Equations | |
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| Separation of Variables | |
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| Fourier Polynomials and Fourier Series | |
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| Fourier Series and Separation of Variables | |
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| Bessel and Legendre Equations | |
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| Series Solutions of Differential Equations | |
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| Expansions About Ordinary Points | |
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| Legendre Polynomials | |
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| Expansions about Singular Points | |
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| Bessel Functions | |
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| Reduction of Order | |
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| Projects | |
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| Systems of Linear Differential Equations | |
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| Existence and Uniqueness of Solutions | |
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| Matrix Algebra | |
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| Diagonalizable Systems with Constant Coefficients | |
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| Non-Diagonalizable Systems with Constant Coefficients | |
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| Qualitative Analysis | |
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| Variation of Parameters | |
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| Outlook on the Theory: Matrix Exponentials and the Jordan Normal Form | |
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| Background | |
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| Tables | |
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| Hints and Solutions for Selected Problems | |
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| Activities | |
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| Bibliography | |
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| Index | |