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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 | |