Preface | p. vii |

Introduction | p. 1 |

Some Basic Mathematical Models; Direction Fields | p. 1 |

Solutions of Some Differential Equations | p. 9 |

Classification of Differential Equations | p. 17 |

Historical Remarks | p. 23 |

First Order Differential Equations | p. 29 |

Linear Equations with Variable Coefficients | p. 29 |

Separable Equations | p. 40 |

Modeling with First Order Equations | p. 47 |

Differences Between Linear and Nonlinear Equations | p. 64 |

Autonomous Equations and Population Dynamics | p. 74 |

Exact Equations and Integrating Factors | p. 89 |

Numerical Approximations: Euler's Method | p. 96 |

The Existence and Uniqueness Theorem | p. 105 |

First Order Difference Equations | p. 115 |

Second Order Linear Equations | p. 129 |

Homogeneous Equations with Constant Coefficients | p. 129 |

Fundamental Solutions of Linear Homogeneous Equations | p. 137 |

Linear Independence and the Wronskian | p. 147 |

Complex Roots of the Characteristic Equation | p. 153 |

Repeated Roots; Reduction of Order | p. 160 |

Nonhomogeneous Equations; Method of Undetermined Coefficients | p. 169 |

Variation of Parameters | p. 179 |

Mechanical and Electrical Vibrations | p. 186 |

Forced Vibrations | p. 200 |

Higher Order Linear Equations | p. 209 |

General Theory of nth Order Linear Equations | p. 209 |

Homogeneous Equations with Constant Coeffients | p. 214 |

The Method of Undetermined Coefficients | p. 222 |

The Method of Variation of Parameters | p. 226 |

Series Solutions of Second Order Linear Equations | p. 231 |

Review of Power Series | p. 231 |

Series Solutions near an Ordinary Point, Part I | p. 238 |

Series Solutions near an Ordinary Point, Part II | p. 249 |

Regular Singular Points | p. 255 |

Euler Equations | p. 260 |

Series Solutions near a Regular Singular Point, Part I | p. 267 |

Series Solutions near a Regular Singular Point, Part II | p. 272 |

Bessel's Equation | p. 280 |

The Laplace Transform | p. 293 |

Definition of the Laplace Transform | p. 293 |

Solution of Initial Value Problems | p. 299 |

Step Functions | p. 310 |

Differential Equations with Discontinuous Forcing Functions | p. 317 |

Impulse Functions | p. 324 |

The Convolution Integral | p. 330 |

Systems of First Order Linear Equations | p. 339 |

Introduction | p. 339 |

Review of Matrices | p. 348 |

Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors | p. 357 |

Basic Theory of Systems of First Order Linear Equations | p. 368 |

Homogeneous Linear Systems with Constant Coefficients | p. 373 |

Complex Eigenvalues | p. 384 |

Fundamental Matrices | p. 393 |

Repeated Eigenvalues | p. 401 |

Nonhomogeneous Linear Systems | p. 411 |

Numerical Methods | p. 419 |

The Euler or Tangent Line Method | p. 419 |

Improvements on the Euler Method | p. 430 |

The Runge-Kutta Method | p. 435 |

Multistep Methods | p. 439 |

More on Errors; Stability | p. 445 |

Systems of First Order Equations | p. 455 |

Nonlinear Differential Equations and Stability | p. 459 |

The Phase Plane; Linear Systems | p. 459 |

Autonomous Systems and Stability | p. 471 |

Almost Linear Systems | p. 479 |

Competing Species | p. 491 |

Predator-Prey Equations | p. 503 |

Liapunov's Second Method | p. 511 |

Periodic Solutions and Limit Cycles | p. 521 |

Chaos and Strange Attractors; the Lorenz Equations | p. 532 |

Partial Differential Equations and Fourier Series | p. 541 |

Two-Point Boundary Valve Problems | p. 541 |

Fourier Series | p. 547 |

The Fourier Convergence Theorem | p. 558 |

Even and Odd Functions | p. 564 |

Separation of Variables; Heat Conduction in a Rod | p. 573 |

Other Heat Conduction Problems | p. 581 |

The Wave Equation; Vibrations of an Elastic String | p. 591 |

Laplace's Equation | p. 604 |

Derivation of the Heat Conduction Equation | p. 614 |

Derivation of the Wave Equation | p. 617 |

Boundary Value Problems and Sturm-Liouville Theory | p. 621 |

The Occurrence of Two Point Boundary Value Problems | p. 621 |

Sturm-Liouville Boundary Value Problems | p. 629 |

Nonhomogeneous Boundary Value Problems | p. 641 |

Singular Sturm-Liouville Problems | p. 656 |

Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion | p. 663 |

Series of Orthogonal Functions: Mean Convergence | p. 669 |

Answers to Problems | p. 679 |

Index | p. 737 |

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