Elementary Differential Equations and Boundary Value Problems

This edition, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between.  The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for reading the book is a working knowledge of calculus, gained from a normal two- or three-semester course sequence or its equivalent.  Some familiarity with matrices will also be helpful in the chapters on systems of differential equations

Table of Contents:
Preface  Chapter 1 Introduction 1 1.1  Some Basic Mathematical Models; Direction Fields  1.2  Solutions of Some Differential Equations  1.3  Classification of Differential Equations  1.4  Historical Remarks Chapter 2 First Order Differential Equations  2.1  Linear Equations; Method of Integrating Factors  2.2 Separable Equations  2.3 Modeling with First Order Equations  2.4 Differences Between Linear and Nonlinear Equations 2.5 Autonomous Equations and Population Dynamics 2.6 Exact Equations and Integrating Factors  2.7 Numerical Approximations: Euler's Method  2.8 The Existence and Uniqueness Theorem 2.9 First Order Difference Equations  Chapter 3 Second Order Linear Equations 135 3.1 Homogeneous Equations with Constant Coefficients  3.2 Fundamental Solutions of Linear Homogeneous Equations; The Wronskian  3.3 Complex Roots of the Characteristic Equation  3.4 Repeated Roots; Reduction of Order 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients  3.6 Variation of Parameters 3.7 Mechanical and Electrical Vibrations 3.8 Forced Vibrations Chapter 4 Higher Order Linear Equations  4.1 General Theory of nth Order Linear Equations 4.2 Homogeneous Equations with Constant Coefficients  4.3 The Method of Undetermined Coefficients  4.4 The Method of Variation of Parameters Chapter 5 Series Solutions of Second Order Linear Equations  5.1 Review of Power Series  5.2 Series Solutions Near an Ordinary Point, Part I  5.3 Series Solutions Near an Ordinary Point, Part II  5.4 Euler Equations; Regular Singular Points  5.5 Series Solutions Near a Regular Singular Point, Part I  5.6 Series Solutions Near a Regular Singular Point, Part II  5.7  Bessel's Equation  Chapter 6 The Laplace Transform 6.1 Definition of the Laplace Transform  6.2 Solution of Initial Value Problems  6.3 Step Functions  6.4 Differential Equations with Discontinuous Forcing Functions  6.5 Impulse Functions  6.6 The Convolution Integral  Chapter 7 Systems of First Order Linear Equations  7.1 Introduction  7.2 Review of Matrices  7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors  7.4 Basic Theory of Systems of First Order Linear Equations 7.5 Homogeneous Linear Systems with Constant Coefficients?  7.6 Complex Eigenvalues  7.7 Fundamental Matrices  7.8 Repeated Eigenvalues  7.9 Nonhomogeneous Linear Systems  Chapter 8 Numerical Methods 8.1 The Euler or Tangent Line Method 8.2 Improvements on the Euler Method  8.3 The Runge-Kutta Method 8.4 Multistep Methods  8.5 More on Errors; Stability  8.6 Systems of First Order Equations Chapter 9 Nonlinear Differential Equations and Stability 9.1 The Phase Plane: Linear Systems  9.2 Autonomous Systems and Stability  9.3 Locally Linear Systems 9.4 Competing Species 9.5 Predator-Prey Equations  9.6 Liapunov's Second Method  9.7 Periodic Solutions and Limit Cycles  9.8 Chaos and Strange Attractors: The Lorenz Equations  Chapter10 Partial Differential Equations and Fourier Series 10.1 Two-Point Boundary Value Problems 10.2 Fourier Series  10.3 The Fourier Convergence Theorem 10.4 Even and Odd Functions  10.5 Separation of Variables; Heat Conduction in a Rod  10.6 Other Heat Conduction Problems  10.7 The Wave Equation: Vibrations of an Elastic String 10.8 Laplace's Equation  Appendix A Derivation of the Heat Conduction Equation  Appendix B Derivation of the Wave Equation  Chapter 11 Boundary Value Problems and Sturm-Liouville Theory 11.1 The Occurrence of Two-Point Boundary Value Problems 11.2 Sturm-Liouville Boundary Value Problems  11.3 Nonhomogeneous Boundary Value Problems  11.4 Singular Sturm-Liouville Problems  11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion  11.6 Series of Orthogonal Functions: Mean Convergence  Answers to Problems  Index