Elementary Differential Equations with Boundary Value Problems

Preface 1 First-Order Differential Equations 1.1 Differential Equations and Mathematical Models 1.2 Integrals as General and Particular Solutions 1.3 Slope Fields and Solution Curves 1.4 Separable Equations and Applications 1.5 Linear First-Order Equations 1.6 Substitution Methods and Exact Equations 1.7 Population Models 1.8 Acceleration-Velocity Models 2 Linear Equations of Higher Order 2.1 Introduction: Second-Order Linear Equations 2.2 General Solutions of Linear Equations 2.3 Homogeneous Equations with Constant Coefficients 2.4 Mechanical Vibrations 2.5 Nonhomogeneous Equations and Undetermined Coefficients 2.6 Forced Oscillations and Resonance 2.7 Electrical Circuits 2.8 Endpoint Problems and Eigenvalues 3 Power Series Methods 3.1 Introduction and Review of Power Series 3.2 Series Solutions Near Ordinary Points 3.3 Regular Singular Points 3.4 Method of Frobenius: The Exceptional Cases 3.5 Bessel's Equation 3.6 Applications of Bessel Functions 4 LaplaceTransform Methods 4.1 Laplace Transforms and Inverse Transforms 4.2 Transformation of Initial Value Problems 4.3 Translation and Partial Fractions 4.4 Derivatives, Integrals, and Products of Transforms 4.5 Periodic and Piecewise Continuous Input Functions 4.6 Impulses and Delta Functions 5 Linear Systems of Differential Equations 5.1 First-Order Systems and Applications 5.2 The Method of Elimination 5.3 Matrices and Linear Systems 5.4 The Eigenvalue Method for Homogeneous Systems 5.5 Second-Order Systems and Mechanical Applications 5.6 Multiple Eigenvalue Solutions 5.7 Matrix Exponentials and Linear Systems 5.8 Nonhomogeneous Linear Systems 6 Numerical Methods 6.1 Numerical Approximation: Euler's Method 6.2 A Closer Look at the Euler Method 6.3 The Runge-Kutta Method 6.4 Numerical Methods for Systems 7 Nonlinear Systems and Phenomena 7.1 Equilibrium Solutions and Stability 7.2 Stability and the Phase Plane 7.3 Linear and Almost Linear Systems 7.4 Ecological Models: Predators and Competitors 7.5 Nonlinear Mechanical Systems 7.6 Chaos in Dynamical Systems 8 Eigenvalues and Boundary Value Problems 8.1 Sturm-Liouville Problems and Eigenfunction Expansions 8.2 Applications of Eigenfunction Series 8.3 Steady Periodic Solutions and Natural Frequencies 8.4 Cylindrical Coordinate Problems 8.5 Higher-Dimensional Phenomena 9 Fourier Series Methods 9.1 Periodic Functions and Trigonometric Series 9.2 General Fourier Series and Convergence 9.3 Fourier Sine and Cosine Series 9.4 Applications of Fourier Series 9.5 Heat Conduction and Separation of Variables 9.6 Vibrating Strings and the One-Dimensional Wave Equation 9.7 Steady-State Temperature and Laplace's Equation Appendix: Existence and Uniqueness of Solutions Answers to Selected Problems Index I-1