Introduction to Ordinary Differential Equations

PrefaceI Introduction to Differential Equations 1.1 Introduction 1.2 Separable Equations 1.3 Exact Equations 1.4 First-Order Linear Equations 1.5 Orthogonal Trajectories 1.6 Decay and Mixing Problems 1.7 Population Growth 1.8 An Economic Model 1.9 Cooling: The Rate of a Chemical Reaction 1.10 Two Special Types of Second-Order Equations 1.11 Falling Bodies ReferencesII Linear Differential Equations 2.1 Introduction 2.2 Linear Dependence 2.3 Wronskians 2.4 Polynomial Operators 2.5 Complex Solutions 2.6 Equations with Constant Coefficients 2.7 Cauchy-Euler Equations 2.8 Nonhomogeneous Equations 2.9 The Method of Undetermined Coefficients 2.10 Variation of Parameters 2.11 Simple Harmonic Motion 2.12 Electric Circuits 2.13 Theory of Linear Equations ReferencesIII Series Solutions 3.1 Power Series 3.2 Taylor Series 3.3 Ordinary Points 3.4 Regular Singular Points 3.5 The Case of Equal Exponents 3.6 The Case When the Exponents Differ by an Integer 3.7 The Point at Infinity 3.8 Convergence of the Series ReferencesIV Bessel Functions 4.1 The Gamma Function 4.2 Bessel's Equation 4.3 Bessel Functions of the Second and Third Kinds 4.4 Properties of Bessel Functions 4.5 Modified Bessel Functions 4.6 Other Forms for Bessel's Equation ReferencesV Orthogonal Polynomials 5.1 Orthogonal Functions 5.2 An Existence Theorem for Orthogonal Polynomials 5.3 Properties of Orthogonal Polynomials 5.4 Generating Functions 5.5 Legendre Polynomials 5.6 Properties of Legendre Polynomials 5.7 Orthogonality 5.8 Legendre's Differential Equation 5.9 Tchebycheff Polynomials 5.10 Other Sets of Orthogonal Polynomials Table of Orthogonal Polynomials ReferencesVI Eigenvalue Problems 6.1 Introduction 6.2 Self-Adjoint Problems 6.3 Some Special Cases 6.4 Singular Problems 6.5 Some Important Singular Problems ReferencesVII Fourier Series 7.1 Introduction 7.2 Examples of Fourier Series 7.3 Types of Convergence 7.4 Convergence in the Mean 7.5 Complete Orthogonal Sets 7.6 Pointwise Convergence of the Trigonometric Series 7.7 The Sine and Cosine Series 7.8 Other Fourier Series ReferencesVIII Systems of Differential Equations 8.1 Introduction 8.2 First-Order Systems 8.3 Linear Systems with Constant Coefficients 8.4 Mechanical Systems 8.5 Electric Circuits 8.6 Some Problems from Biology ReferencesIX Laplace Transforms 9.1 The Laplace Transform 9.2 Conditions for the Existence of the Laplace Transform 9.3 Properties of Laplace Transforms 9.4 Inverse Transforms 9.5 Applications to Differential Equations ReferencesX Partial Differential Equations and Boundary Value Problems 10.1 Introduction 10.2 The Heat Equation 10.3 The Method of Separation of Variables 10.4 Steady-State Heat Flow 10.5 The Vibrating String 10.6 The Solution of the Problem of the Vibrating String 10.7 The Laplacian in Other Coordinate Systems 10.8 A Problem in Cylindrical Coordinates 10.9 A Problem in Spherical Coordinates 10.10 Double Fourier Series ReferencesXI The Phase Plane 11.1 Introduction 11.2 Stability 11.3 The Method of Liapunov 11.4 Perturbed Linear Systems 11.5 Periodic Solutions ReferencesXII Existence and Uniqueness of Solutions 12.1 Preliminaries 12.2 Successive Approximations 12.3 Vector Functions 12.4 First-Order Systems ReferencesAppendix A1 Determinants A2 Properties of Determinants A3 Cofactors A4 Cramer's RuleAnswers to Selected ExercisesSubject Index