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| Preface | |
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| Introduction | |
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| Some Basic Mathematical Models; Direction Fields | |
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| Solutions of Some Differential Equations | |
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| Classification of Differential Equations | |
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| Historical Remarks | |
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| First Order Differential Equations | |
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| Linear Equations with Variable Coefficients | |
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| Separable Equations | |
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| Modeling with First Order Equations | |
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| Differences Between Linear and Nonlinear Equations | |
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| Autonomous Equations and Population Dynamics | |
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| Exact Equations and Integrating Factors | |
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| Numerical Approximations: Euler's Method | |
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| The Existence and Uniqueness Theorem | |
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| First Order Difference Equations | |
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| Second Order Linear Equations | |
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| Homogeneous Equations with Constant Coefficients | |
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| Fundamental Solutions of Linear Homogeneous Equations | |
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| Linear Independence and the Wronskian | |
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| Complex Roots of the Characteristic Equation | |
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| Repeated Roots; Reduction of Order | |
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| Nonhomogeneous Equations; Method of Undetermined Coefficients | |
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| Variation of Parameters | |
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| Mechanical and Electrical Vibrations | |
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| Forced Vibrations | |
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| Higher Order Linear Equations | |
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| General Theory of nth Order Linear Equations | |
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| Homogeneous Equations with Constant Coeffients | |
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| The Method of Undetermined Coefficients | |
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| The Method of Variation of Parameters | |
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| Series Solutions of Second Order Linear Equations | |
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| Review of Power Series | |
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| Series Solutions near an Ordinary Point, Part I | |
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| Series Solutions near an Ordinary Point, Part II | |
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| Regular Singular Points | |
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| Euler Equations | |
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| Series Solutions near a Regular Singular Point, Part I | |
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| Series Solutions near a Regular Singular Point, Part II | |
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| Bessel's Equation | |
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| The Laplace Transform | |
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| Definition of the Laplace Transform | |
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| Solution of Initial Value Problems | |
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| Step Functions | |
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| Differential Equations with Discontinuous Forcing Functions | |
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| Impulse Functions | |
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| The Convolution Integral | |
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| Systems of First Order Linear Equations | |
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| Introduction | |
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| Review of Matrices | |
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| Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors | |
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| Basic Theory of Systems of First Order Linear Equations | |
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| Homogeneous Linear Systems with Constant Coefficients | |
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| Complex Eigenvalues | |
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| Fundamental Matrices | |
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| Repeated Eigenvalues | |
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| Nonhomogeneous Linear Systems | |
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| Numerical Methods | |
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| The Euler or Tangent Line Method | |
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| Improvements on the Euler Method | |
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| The Runge-Kutta Method | |
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| Multistep Methods | |
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| More on Errors; Stability | |
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| Systems of First Order Equations | |
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| Nonlinear Differential Equations and Stability | |
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| The Phase Plane; Linear Systems | |
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| Autonomous Systems and Stability | |
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| Almost Linear Systems | |
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| Competing Species | |
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| Predator-Prey Equations | |
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| Liapunov's Second Method | |
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| Periodic Solutions and Limit Cycles | |
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| Chaos and Strange Attractors; the Lorenz Equations | |
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| Partial Differential Equations and Fourier Series | |
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| Two-Point Boundary Valve Problems | |
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| Fourier Series | |
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| The Fourier Convergence Theorem | |
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| Even and Odd Functions | |
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| Separation of Variables; Heat Conduction in a Rod | |
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| Other Heat Conduction Problems | |
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| The Wave Equation; Vibrations of an Elastic String | |
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| Laplace's Equation | |
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| Derivation of the Heat Conduction Equation | |
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| Derivation of the Wave Equation | |
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| Boundary Value Problems and Sturm-Liouville Theory | |
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| The Occurrence of Two Point Boundary Value Problems | |
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| Sturm-Liouville Boundary Value Problems | |
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| Nonhomogeneous Boundary Value Problems | |
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| Singular Sturm-Liouville Problems | |
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| Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion | |
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| Series of Orthogonal Functions: Mean Convergence | |
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| Answers to Problems | |
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| Index | |