内容

目次

0. Complex Numbers. Introduction. The Cartesian and Exponential Forms.Roots of Polynomial Equations and Numbers. Matrix Notation and Terminology.The Solution of Simultaneous Equations. The Algebra of Matrices. MatrixMultiplication. The Inverse of a Matrix. The Computation of A-1.Determinants. Linear Independence. 1. First-Order Differential Equations.Preliminaries. Definitions. The First-Order Linear Equation. Applications ofFirst-Order Linear Equations. Nonlinear Equations of First Order. 2. LinearSystems. Introduction. Eigenvalues and Eigenvectors. First-Order Systems.Solution and Fundamental Solution Matrices. Some Fundamental Theorems.Solutions of Nonhomogeneous Systems. Nonhomogeneous Initial-Value Problems.Fundamental Solution Matrices. 3. Second-Order Linear Equations.Introduction. Sectionally Continuous Functions. Linear DifferentialOperators. Linear Independence and the Wronskian. The NonhomogeneousEquation. Constant Coefficient Equations. Spring-Mass Systems in Free Motion.The Electric Circuit. Undetermined Coefficients. The Spring-Mass System:Forced Motion. The Cauchy-Euler Equation. Variation of Parameters. 4. HigherOrder Equations. Introduction. The Homogeneous Equation. The NonhomogeneousEquation. Companion Systems. Homogeneous Companion Systems. Variation ofParameters. 5. The Laplace Transform. Introduction. Preliminaries. GeneralProperties of the Laplace Transform. Sectionally Continuous Functions.Laplace Transforms of Periodic Functions. The Inverse Laplace Transform.Partial Fractions. The Convolution Theorem. The Solution of Initial-ValueProblems. The Laplace Transform of Systems. Tables of Transforms. 6. SeriesMethods. Introduction. Analytic Functions. Taylor Series of AnalyticFunctions. Power Series Solutions. Legendre's Equations. Three ImportantExamples. Bessel's Equation. The Wronskian Method. The Frobenius Method. 7.Numerical Methods. Introduction. Direction Fields. Notational Conventions.The Euler Method. Heun's Method. Taylor Series Methods. Runge-Kutta Methods.Multivariable Methods. Higher Order Equations. 8. Boundary-Value Problems.Introduction. Separation of Variables. Fourier Series Expansions. The WaveEquation. The One-Dimensional Heat Equation. The Laplace Equation. APotential about a Spherical Surface.