Content: Preface 1 What is a Differential Equation? 1.1 Introductory Remarks 1.2 The Nature of Solutions 1.3 Separable Equations 1.4 First-Order Linear Equations 1.5 Exact Equations 1.6 Orthogonal Trajectories and Families of Curves 1.7 Homogeneous Equations 1.8 Integrating Factors 1.9 Reduction of Order 1.9.1 Dependent Variable Missing 1.9.2 Independent Variable Missing 1.10 The Hanging Chain and Pursuit Curves 1.10.1 The Hanging Chain 1.10.2 Pursuit Curves 1.11 Electrical Circuits Anatomy of an Application: The Design of a Dialysis Machine Problems for Review and Discovery 2 Second-Order Linear Equations 2.1 Second-Order Linear Equations with Constant Coefficients 2.2 The Method of Undetermined Coefficients 2.3 The Method of Variation of Parameters 2.4 The Use of a Known Solution to Find Another 2.5 Vibrations and Oscillations 2.5.1 Undamped Simple Harmonic Motion 2.5.2 Damped Vibrations 2.5.3 Forced Vibrations 2.5.4 A Few Remarks About Electricity 2.6 Newton's Law of Gravitation and Kepler's Laws 2.6.1 Kepler's Second Law 2.6.2 Kepler's First Law 2.6.3 Kepler's Third Law 2.7 Higher Order Linear Equations, Coupled Harmonic Oscillators Historical Note: Euler Anatomy of an Application: Bessel Functions and the Vibrating Membrane Problems for Review and Discovery 3 Qualitative Properties and Theoretical Aspects 3.1 Review of Linear Algebra 3.1.1 Vector Spaces 3.1.2 The Concept Linear Independence 3.1.3 Bases 3.1.4 Inner Product Spaces 3.1.5 Linear Transformations and Matrices 3.1.6 Eigenvalues and Eigenvectors 3.2 A Bit of Theory 3.3 Picard's Existence and Uniqueness Theorem 3.3.1 The Form of a Differential Equation 3.3.2 Picard's Iteration Technique 3.3.3 Some Illustrative Examples 3.3.4 Estimation of the Picard Iterates 3.4 Oscillations and the Sturm Separation Theorem 3.5 The Sturm Comparison Theorem Anatomy of an Application: The Green's Function Problems for Review and Discovery 4 Power Series Solutions and Special Functions 4.1 Introduction and Review of Power Series 4.1.1 Review of Power Series 4.2 Series Solutions of First-Order Differential Equations 4.3 Second-Order Linear Equations: Ordinary Points 4.4 Regular Singular Points 4.5 More on Regular Singular Points 4.6 Gauss's Hypergeometric Equation Historical Note: Gauss Historical Note: Abel Anatomy of an Application: Steady-State Temperature in a Ball Problems for Review and Discovery 5 Fourier Series: Basic Concepts 5.1 Fourier Coefficients 5.2 Some Remarks about Convergence 5.3 Even and Odd Functions: Cosine and Sine Series 5.4 Fourier Series on Arbitrary Intervals 5.5 Orthogonal Functions Historical Note: Riemann Anatomy of an Application: Introduction to the Fourier Transform Problems for Review and Discovery 6 Partial Differential Equations and Boundary Value Problems 6.1 Introduction and Historical Remarks 6.2 Eigenvalues, Eigenfunctions, and the Vibrating String 6.2.1 Boundary Value Problems 6.2.2 Derivation of the Wave Equation 6.2.3 Solution of the Wave Equation 6.3 The Heat Equation 6.4 The Dirichlet Problem for a Disc 6.4.1 The Poisson Integral 6.5 Sturm-Liouville Problems Historical Note: Fourier Historical Note: Dirichlet Anatomy of an Application: Some Ideas from Quantum Mechanics Problems for Review and Discovery 7 Laplace Transforms 7.1 Introduction 7.2 Applications to Differential Equations 7.3 Derivatives and Integrals of Laplace Transforms 7.4 Convolutions 7.3.1 Abel's Mechanical Problem 7.5 The Unit Step and Impulse Functions Historical Note: Laplace Anatomy of an Application: Flow Initiated by an Impulsively-Started Flat Plate Problems for Review and Discovery 8 The Calculus of Variations 8.1 Introductory Remarks 8.2 Euler's Equation 8.3 Isoperimetric Problems and the Like 8.3.1 Lagrange Multipliers 8.3.2 Integral Side Conditions 8.3.3 Finite Side Conditions Historical Note: Newton Anatomy of an Application: Hamilton's Principle and its Implications Problems for Review and Discovery 9 Numerical Methods 9.1 Introductory Remarks 9.2 The Method of Euler 9.3 The Error Term 9.4 An Improved Euler Method 9.5 The Runge-Kutta Method Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations Problems for Review and Discovery 10 Systems of First-Order Equations 10.1 Introductory Remarks 10.2 Linear Systems 10.3 Homogeneous Linear Systems with Constant Coefficients 10.4 Nonlinear Systems: Volterra's Predator-Prey Equations Anatomy of an Application: Solution of Systems with Matrices and Exponentials Problems for Review and Discovery 11 The Nonlinear Theory 11.1 Some Motivating Examples 11.2 Specializing Down 11.3 Types of Critical Points: Stability 11.4 Critical Points and Stability for Linear Systems 11.5 Stability by Liapunov's Direct Method 11.6 Simple Critical Points of Nonlinear Systems 11.7 Nonlinear Mechanics: Conservative Systems 11.8 Periodic Solutions: The Poincare-Bendixson Theorem Historical Note: Poincare Anatomy of an Application: Mechanical Analysis of a Block on a Spring Problems for Review and Discovery 12 Dynamical Systems 12.1 Flows 12.1.1 Dynamical Systems 12.1.2 Stable and Unstable Fixed Points 12.1.3 Linear Dynamics in the Plane 12.2 Some Ideas from Topology 12.2.1 Open and Closed Sets 12.2.2 The Idea of Connectedness 12.2.3 Closed Curves in the Plane 12.3 Planar Autonomous Systems 12.3.1 Ingredients of the Proof of Poincare-Bendixson Anatomy of an Application: Lagrange's Equations Problems for Review and Discovery Bibliography