Preface | p. VII |

What is a Differential Equation? | p. 1 |

Introductory Remarks | p. 2 |

The Nature of Solutions | p. 4 |

Separable Equations | p. 10 |

First-Order Linear Equations | p. 13 |

Exact Equations | p. 17 |

Orthogonal Trajectories and Families of Curves | p. 22 |

Homogeneous Equations | p. 26 |

Integrating Factors | p. 29 |

Reduction of Order | p. 33 |

Dependent Variable Missing | p. 33 |

Independent Variable Missing | p. 35 |

The Hanging Chain and Pursuit Curves | p. 38 |

The Hanging Chain | p. 38 |

Pursuit Curves | p. 42 |

Electrical Circuits | p. 45 |

Anatomy of an Application: The Design of a Dialysis Machine | p. 49 |

Problems for Review and Discovery | p. 53 |

Second-Order Linear Equations | p. 57 |

Second-Order Linear Equations with Constant Coefficients | p. 58 |

The Method of Undetermined Coefficients | p. 63 |

The Method of Variation of Parameters | p. 67 |

The Use of a Known Solution to Find Another | p. 71 |

Vibrations and Oscillations | p. 75 |

Undamped Simple Harmonic Motion | p. 75 |

Damped Vibrations | p. 77 |

Forced Vibrations | p. 80 |

A Few Remarks about Electricity | p. 82 |

Newton's Law of Gravitation and Kepler's Laws | p. 84 |

Kepler's Second Law | p. 86 |

Kepler's First Law | p. 87 |

Kepler's Third Law | p. 89 |

Higher Order Linear Equations, Coupled Harmonic Oscillators | p. 93 |

Historical Note: Euler | p. 99 |

Anatomy of an Application: Bessel Functions and the Vibrating Membrane | p. 101 |

Problems for Review and Discovery | p. 105 |

Qualitative Properties and Theoretical Aspects | p. 109 |

Review of Linear Algebra | p. 110 |

Vector Spaces | p. 110 |

The Concept of Linear Independence | p. 111 |

Bases | p. 113 |

Inner Product Spaces | p. 114 |

Linear Transformations and Matrices | p. 115 |

Eigenvalues and Eigenvectors | p. 117 |

A Bit of Theory | p. 119 |

Picard's Existence and Uniqueness Theorem | p. 125 |

The Form of a Differential Equation | p. 125 |

Picard's Iteration Technique | p. 126 |

Some Illustrative Examples | p. 127 |

Estimation of the Picard Iterates | p. 129 |

Oscillations and the Sturm Separation Theorem | p. 130 |

The Sturm Comparison Theorem | p. 138 |

Anatomy of an Application: The Green's Function | p. 142 |

Problems for Review and Discovery | p. 146 |

Power Series Solutions and Special Functions | p. 149 |

Introduction and Review of Power Series | p. 150 |

Review of Power Series | p. 150 |

Series Solutions of First-Order Differential Equations | p. 159 |

Second-Order Linear Equations: Ordinary Points | p. 164 |

Regular Singular Points | p. 171 |

More on Regular Singular Points | p. 177 |

Gauss's Hypergeometric Equation | p. 184 |

Historical Note: Gauss | p. 189 |

Historical Note: Abel | p. 190 |

Anatomy of an Application: Steady-State Temperature in a Ball | p. 192 |

Problems for Review and Discovery | p. 194 |

Fourier Series: Basic Concepts | p. 197 |

Fourier Coefficients | p. 198 |

Some Remarks about Convergence | p. 207 |

Even and Odd Functions: Cosine and Sine Series | p. 211 |

Fourier Series on Arbitrary Intervals | p. 218 |

Orthogonal Functions | p. 221 |

Historical Note: Riemann | p. 225 |

Anatomy of an Application: Introduction to the Fourier Transform | p. 227 |

Problems for Review and Discovery | p. 236 |

Partial Differential Equations and Boundary Value Problems | p. 239 |

Introduction and Historical Remarks | p. 240 |

Eigenvalues, Eigenfunctions, and the Vibrating String | p. 243 |

Boundary Value Problems | p. 243 |

Derivation of the Wave Equation | p. 244 |

Solution of the Wave Equation | p. 246 |

The Heat Equation | p. 251 |

The Dirichlet Problem for a Disc | p. 256 |

The Poisson Integral | p. 259 |

Sturm-Liouville Problems | p. 262 |

Historical Note: Fourier | p. 267 |

Historical Note: Dirichlet | p. 268 |

Anatomy of an Application: Some Ideas from Quantum Mechanics | p. 270 |

Problems for Review and Discovery | p. 273 |

Laplace Transforms | p. 277 |

Introduction | p. 278 |

Applications to Differential Equations | p. 280 |

Derivatives and Integrals of Laplace Transforms | p. 285 |

Convolutions | p. 291 |

Abel's Mechanical Problem | p. 293 |

The Unit Step and Impulse Functions | p. 298 |

Historical Note: Laplace | p. 305 |

Anatomy of an Application: Flow Initiated by an Impulsively Started Flat Plate | p. 306 |

Problems for Review and Discovery | p. 309 |

The Calculus of Variations | p. 315 |

Introductory Remarks | p. 316 |

Euler's Equation | p. 319 |

Isoperimetric Problems and the Like | p. 327 |

Lagrange Multipliers | p. 328 |

Integral Side Conditions | p. 329 |

Finite Side Conditions | p. 333 |

Historical Note: Newton | p. 338 |

Anatomy of an Application: Hamilton's Principle and its Implications | p. 340 |

Problems for Review and Discovery | p. 344 |

Numerical Methods | p. 347 |

Introductory Remarks | p. 348 |

The Method of Euler | p. 349 |

The Error Term | p. 353 |

An Improved Euler Method | p. 357 |

The Runge-Kutta Method | p. 360 |

Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations | p. 365 |

Problems for Review and Discovery | p. 368 |

Systems of First-Order Equations | p. 371 |

Introductory Remarks | p. 372 |

Linear Systems | p. 374 |

Homogeneous Linear Systems with Constant Coefficients | p. 382 |

Nonlinear Systems: Volterra's Predator-Prey Equations | p. 389 |

Anatomy of an Application: Solution of Systems with Matrices and Exponentials | p. 395 |

Problems for Review and Discovery | p. 400 |

The Nonlinear Theory | p. 403 |

Some Motivating Examples | p. 404 |

Specializing Down | p. 404 |

Types of Critical Points: Stability | p. 409 |

Critical Points and Stability for Linear Systems | p. 417 |

Stability by Liapunov's Direct Method | p. 427 |

Simple Critical Points of Nonlinear Systems | p. 432 |

Nonlinear Mechanics: Conservative Systems | p. 439 |

Periodic Solutions: The Poincare-Bendixson Theorem | p. 444 |

Historical Note: Poincare | p. 452 |

Anatomy of an Application: Mechanical Analysis of a Block on a Spring | p. 454 |

Problems for Review and Discovery | p. 457 |

Dynamical Systems | p. 461 |

Flows | p. 462 |

Dynamical Systems | p. 464 |

Stable and Unstable Fixed Points | p. 466 |

Linear Dynamics in the Plane | p. 468 |

Some Ideas from Topology | p. 475 |

Open and Closed Sets | p. 475 |

The Idea of Connectedness | p. 476 |

Closed Curves in the Plane | p. 478 |

Planar Autonomous Systems | p. 480 |

Ingredients of the Proof of Poincare-Bendixson | p. 480 |

Anatomy of an Application: Lagrange's Equations | p. 489 |

Problems for Review and Discovery | p. 493 |

Bibliography | p. 495 |

Answers to Odd-Numbered Exercises | p. 497 |

Index | p. 525 |

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