Differential Equations with Applications: (Dover Books on Mathematics)

Coherent introductory text focuses on initial- and boundary-value problems, general properties of linear equations, and differences between linear and nonlinear systems. Answers to most problems.

Table of Contents:
Preface Chap. 1 Basic Concepts 1-1 Introduction 1-2 Classifications and Examples of Differential Equations 1-3 The Motion of a Particle 1-4 The Solution of a Differential Equation 1-5 Initial- and Boundary-value Problems 1-6 The Differential Equation y' = f(x) 1-7 Integrals as Functions of Parameters 1-8 Elementary and Non-elementary Functions 1-9 The Gamma Function Appendix Chap. 2 Special Methods for First-order Equations 2-1 Introduction 2-2 Separation of Variables 2-3 The First-order Linear Differential Equation 2-4 Exact Differential Equations 2-5 Integrating Factors 2-6 Use of Substitutions 2-7 Second-order Equations Reducible to First-order 2-8 Summary Chap. 3 Applications of First-order Equations 3-1 Introduction 3-2 Falling Bodies with Air Resistance 3-3 Motion on a Given Curve 3-4 Linear Motion with Variable Mass 3-5 Newton's Law of Cooling 3-6 Dilution Problems 3-7 Chemical Reactions 3-8 Population Growth 3-9 A Simple Electrical Circuit 3-10 Families of Curves and Orthogonal Trajectories Chap. 4 Existence and Uniqueness and Methods of Approximation 4-1 Introduction 4-2 The Direction Field 4-3 Existence and Uniqueness of Solutions 4-4 The Picard Method 4-5 The Cauchy-Euler Method 4-6 Taylor Series 4-7 Existence and Uniqueness Theorems for Systems of Equations and Higher-order Equations 4-8 Existence and Uniqueness Theorems for Linear Equations Appendix: Proof of existence and uniqueness for Linear Equations Chap. 5 Linear Differential Equations 5-1 Introduction 5-2 Fundamental Theory of Second-order Linear Equations 5-3 Complex-valued Solutions 5-4 Homogeneous Linear Equations with Constant Coefficients 5-5 Undetermined Coefficients 5-6 Variation of Parameters 5-7 Euler's Equation 5-8 Formulas of Lagrange and Abel 5-9 Linear Equations of the nth Order Chap. 6 Applications of Second-order Linear Differential Equations 6-1 Introduction 6-2 Free Vibrations 6-3 Forced Vibrations 6-4 Electrical Circuits 6-5 The Equations of Planetary Motives Chap. 7 Linear Differential Equations with Variable Coefficients 7-1 Introduction 7-2 Solution by Power Series 7-3 Solution near a Singular Point 7-4 Bessel's Equation 7-5 Hypergeometric Equation 7-6 Legendre's Equation Chap. 8 Systems of Linear Differential Equations 8-1 Introduction 8-2 Some Illustrative Examples 8-3 A Two-degree-of-freedom Vibration 8-4 Vectors and Matrics 8-5 Theory of Systems of Linear Differential Equations 8-6 Homogeneous Linear Systems with Constant Coefficients 8-7 Solution by Matrix Methods Chap. 9 The Laplace Transform 9-1 Introduction 9-2 Improper Integrals 9-3 The Laplace Transform 9-4 Properties of the Laplace Transform 9-5 Solution of Linear Equations with Constant Coefficients 9-6 Product of Transform Functions; Convolutions 9-7 Discontinuous Functions 9-8 Linear Systems Analysis Appendix Chap. 10 Nonlinear Differential Equations 10-1 Introduction 10-2 The Pendulum 10-3 Singularities and the Phase Plane 10-4 Van der Pol's Equation 10-5 Piecewise Linear System 10-6 Liapunov's Second Method Chap. 11 Linear Difference Equations 11-1 Introduction 11-2 First-order Linear Difference Equations 11-3 Second-order Linear Difference Equations 11-4 Homogeneous Linear Difference Equations with Constant Coefficients 11-5 The Nonhomogeneous Equation 11-6 The Vector Space EN 11-7 A Boundary-value Problem 11-8 N Beads on a Tightly Stretched String Chap. 12 Numerical Methods 12-1 Introduction 12-2 The Euler Method 12-3 Error Analysis 12-4 Parasitic Solutions and Stability 12-5 A Second-order Predictor-Corrector Method 12-6 Fourth-order Predictor-Corrector Methods 12-7 Starting Methods and Runge-Kutta Methods 12-8 Higher-order Equations and Systems of Equations Chap. 13 Boundary-value Problems 13-1 Introduction 13-2 Homogenous Boundary-value Problems 13-3 Eigenvalue Problems 13-4 Orthogonal Functions 13-5 Generalized Fourier Series 13-6 Weight Functions 13-7 The Sturm-Liouville Problem 13-8 Theorems on Eigenvalues and Eigenfunctions 13-9 Ordinary Fourier Series 13-10 Fourier-Bessel Series 13-11 Fourier-Legendre Series 13-12 Nonhomogeneous Boundary-value Problems Chap. 14 Partial Differential Equations of Mathematical Physics 14-1 Introduction 14-2 The Vibrating String 14-3 Heat Conduction 14-4 Laplace's Equation 14-5 Theory of Second-order Equations Chap. 15 Further Applications of Partial Differential Equations 15-1 Introduction 15-2 Laplace's Equation in Three Dimensions 15-3 Temperature in an Infinite Cylinder 15-4 Vibrating Membranes Appendix A Infinite Series Appendix B Functions of a Complex Variable References Answers and Hints Index