Differential equations: a systems approach

Abstract

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. Matrix Multiplication. 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 of First-Order Linear Equations. Nonlinear Equations of First Order. 2. Linear Systems. 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 Differential Operators. Linear Independence and the Wronskian. The Nonhomogeneous Equation. 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. Higher Order Equations. Introduction. The Homogeneous Equation. The Nonhomogeneous Equation. Companion Systems. Homogeneous Companion Systems. Variation of Parameters. 5. The Laplace Transform. Introduction. Preliminaries. General Properties 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-Value Problems. The Laplace Transform of Systems. Tables of Transforms. 6. Series Methods. Introduction. Analytic Functions. Taylor Series of Analytic Functions. Power Series Solutions. Legendre's Equations. Three Important Examples. 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 Wave Equation. The One-Dimensional Heat Equation. The Laplace Equation. A Potential about a Spherical Surface.