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| Preface | |
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| Introduction to Differential Equations | |
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| Definitions and Terminology | |
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| Linear First-Order Equations | |
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| Homogeneous Case | |
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| Solution by Separation of Variables | |
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| Nonhomogeneous Case | |
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| Applications of Linear First-Order Equations | |
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| General First-Order Equations | |
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| Separable Equations | |
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| Existence and Uniqueness | |
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| Exact Equations and Ones That Can Be Made Exact | |
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| Additional Applications | |
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| Linear and Nonlinear Equations Contrasted | |
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| Vectors and n-Space | |
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| Geometrical Representation of Arrow Vectors n-Space | |
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| Dot Product, Norm, and Angle for n-Space | |
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| Gauss Elimination | |
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| Span | |
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| Linear Dependence and Independence | |
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| Vector Space | |
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| Bases and Expansions | |
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| Matrices and Linear Algebraic Equations | |
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| Matrices and Matrix Algebra | |
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| The Transpose Matrix | |
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| Determinants | |
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| The Rank of a Matrix | |
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| Inverse Matrix, Cramer's Rule, and Factorization | |
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| Existence and Uniqueness for the System Ax= c | |
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| Vector Transformation (Optional) | |
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| Linear Differential Equations of Second Order and Higher | |
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| The Complex Plane and the Exponential, Trigonometric, and Hyperbolic Functions | |
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| Linear Dependence and Linear Independence of Functions | |
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| Homogeneous Equation | |
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| General Solution | |
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| Homogeneous Equations with Constant Coefficients | |
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| Homogeneous Equations with Nonconstant Coefficients | |
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| Cauchy-Euler Equation | |
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| Solution of Nonhomogeneous Equation | |
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| Applications of Linear Constant-Coefficient Equations | |
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| Linear Harmonic Oscillator | |
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| Free Oscillation | |
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| Linear Harmonic Oscillator | |
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| Phase Plane | |
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| Linear Harmonic Oscillator | |
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| Forced Oscillation | |
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| Additional Applications | |
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| Power Series Solution | |
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| Power Series | |
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| Power Series Solutions | |
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| Introduction to the Singular Case | |
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| The Eigenvalue Problem | |
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| Solution of the Eigenvalue Problem | |
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| The Special Case of Symmetric Matrices | |
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| Differential Equation Boundary Value Problems as Eigenvalue Problems | |
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| Systems of Linear Differential Equations | |
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| Existence, Uniqueness, and General Solution | |
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| Solution by Elimination | |
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| Constant-Coefficient Equations | |
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| Solution of Homogeneous Systems as Eigenvalue Problems | |
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| Diagonalization (Optional) | |
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| Triangularization (Optional) | |
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| Explicit Solution of x' = Ax and the Matrix Exponential Function (Optional) | |
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| Quantitative Methods: The Phase Plane | |
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| The Phase Plane | |
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| Singular Point Analysis | |
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| Additional Applications | |
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| Qualitative Methods: Numerical Solution | |
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| Euler's Method | |
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| Improvements: Runge-Kutta Methods | |
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| Application to Systems and Boundary Value Problems | |
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| Laplace Transform | |
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| Definitions and Calculation of the Transform | |
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| Properties of the Transform | |
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| Application to the Solution of Differential Equations | |
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| Discontinuous Forcing Functions | |
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| Heaviside Step Function | |
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| Additional Properties | |
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| Appendix | |
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| Answers to Selected Exercises | |
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| Index | |