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Complex Numbers | |
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Introduction | |
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The Cartesian and Exponential Forms | |
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Roots of Polynomial Equations and Numbers | |
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Matrix Notation and Terminology | |
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The Solution of Simultaneous Equations | |
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The Algebra of Matrices | |
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Matrix Multiplication | |
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The Inverse of a Matrix | |
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The Computation of A-1 | |
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Determinants | |
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Linear Independence | |
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First-Order Differential Equations | |
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Preliminaries | |
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Definitions | |
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The First-Order Linear Equation | |
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Applications of First-Order Linear Equations | |
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Nonlinear Equations of First Order | |
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Linear Systems | |
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Introduction | |
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Eigenvalues and Eigenvectors | |
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First-Order Systems | |
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Solution and Fundamental Solution Matrices | |
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Some Fundamental Theorems | |
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Solutions of Nonhomogeneous Systems | |
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Nonhomogeneous Initial-Value Problems | |
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Fundamental Solution Matrices | |
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Second-Order Linear Equations | |
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Introduction | |
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Sectionally Continuous Functions | |
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Linear Differential Operators | |
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Linear Independence and the Wronskian | |
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The Nonhomogeneous Equation | |
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Constant Coefficient Equations | |
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Spring-Mass Systems in Free Motion | |
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The Electric Circuit | |
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Undetermined Coefficients | |
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The Spring-Mass System: Forced Motion | |
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The Cauchy-Euler Equation | |
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Variation of Parameters | |
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Higher Order Equations | |
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Introduction | |
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The Homogeneous Equation | |
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The Nonhomogeneous Equation | |
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Companion Systems | |
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Homogeneous Companion Systems | |
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Variation of Parameters | |
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The Laplace Transform | |
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Introduction | |
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Preliminaries | |
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General Properties of the Laplace Transform | |
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Sectionally Continuous Functions | |
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Laplace Transforms of Periodic Functions | |
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The Inverse Laplace Transform | |
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Partial Fractions | |
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The Convolution Theorem | |
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The Solution of Initial-Value Problems | |
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The Laplace Transform of Systems | |
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Tables of Transforms | |
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Series Methods | |
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Introduction | |
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Analytic Functions | |
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Taylor Series of Analytic Functions | |
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Power Series Solutions | |
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Legendre's Equations | |
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Three Important Examples | |
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Bessel's Equation | |
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The Wronskian Method | |
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The Frobenius Method | |
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Numerical Methods | |
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Introduction | |
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Direction Fields | |
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Notational Conventions | |
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The Euler Method | |
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Heun's Method | |
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Taylor Series Methods | |
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Runge-Kutta Methods | |
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Multivariable Methods | |
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Higher Order Equations | |
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Boundary-Value Problems | |
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Introduction | |
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Separation of Variables | |
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Fourier Series Expansions | |
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The Wave Equation | |
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The One-Dimensional Heat Equation | |
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The Laplace Equation | |
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A Potential about a Spherical Surface | |