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| Introduction | |
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| What Is a Differential Equation? | |
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| Applications of Differential Equations | |
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| Approaches to Solving Differential Equations. Reduction to First-Order Systems | |
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| The First-Order Equation y' = f (x, y) | |
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| The Graphical Viewpoint (Direction Fields) | |
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| Numerical Methods | |
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| Analytic Methods of Solution | |
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| Autonomous Equations and Critical Points | |
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| The Dependence of Solutions on Initial Conditions | |
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| Introduction to First-Order Systems | |
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| Solutions to First-Order Systems | |
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| Orbit Crossing and Periodic Solutions | |
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| Numerical Approximations | |
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| Some Qualitative Behavior | |
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| The Pendulum | |
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| Limit Cycles | |
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| Higher-Order Linear Equations | |
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| Introduction | |
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| A Strategy for Solving Linear Homogeneous Equations | |
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| Linear Homogeneous Equations with Constant Coefficients | |
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| NonHomogeneous Equations | |
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| Vibration | |
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| First-Order Systems: Linear Methods | |
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| Matrices, Independence, and Eigenvectors | |
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| Solving 2 x 2 Linear Systems | |
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| The Matrix Exponential Function | |
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| Qualitative Behavior of Linear Systems | |
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| A Couple System of Masses and Springs | |
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| Linearization of 2 x 2 Systems | |
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| Series Methods and Famous Functions | |
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| A Power Series Method | |
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| Famous Functions | |
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| Regular Singular Points and the Method of Frobenius | |
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| The Exceptional Cases | |
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| Bifurcations and Chaos | |
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| Bifurcation | |
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| Flows. A Basic Theorem | |
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| Some Simple Attractors | |
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| The Periodically Driven Pendulum | |
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| Chaos | |
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| The Laplace Transform | |
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| Introduction | |
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| Transforms of Basic Functions | |
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| Solving Linear Homogeneous Equations | |
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| NonHomogeneous Equations and the Convolution | |
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| Discontinuous and Impulsive Forcing Functions | |
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| Laplace Transforms and Systems | |
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| Poles and Qualitative Behavior | |
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| Partial Differential Equations and Fourier Series | |
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| Some General Remarks | |
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| The Heat Equation, Wave Equation, and Laplace's Equation | |
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| The Heat Equation and Initial Condition | |
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| Vector Spaces and Operators | |
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| The Heat Equation Revisited | |
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| Periodic Functions and Fourier Series | |
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| The One-Dimensional Wave Equation | |
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| The Convergence of Series | |
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| The Two-Dimensional Wave Equation | |
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| Laplace's Equation | |
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| The Finite Differences Method | |
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| Finite Differences Approximations | |
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| An Example | |
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| The Heat and Wave Equations | |
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| A Backward Method for the Heat Equation | |
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| Variable Coefficient and NonLinear Examples | |
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| Laplace's Equation | |
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| Stability | |
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| Appendices | |
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| Linear Systems, Matrices, and Determinants | |
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| Solution of Systems of Equations | |
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| Matrices, Inverses, and Determinants | |
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| The Two-Variable Taylor Theorem | |
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| The Existence and Uniqueness Theorem | |
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| Mathematica, Maple, and MatLab | |
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| Graphing | |
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| Field&Solution | |
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| ERGraphical | |
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| ERNumerical | |
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| 2x2Systems | |
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| 2x2Numerical | |
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| Fourier Series | |
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| Answers and Hints to Odd-Numbered Exercises | |
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