Preface | p. ix |

Introduction to Differential Equations | p. 1 |

Differential Equation Models | p. 2 |

The Derivative | p. 7 |

Integration | p. 10 |

First-Order Equations | p. 18 |

Differential Equations and Solutions | p. 18 |

Solutions to Separable Equations | p. 31 |

Models of Motion | p. 44 |

Linear Equations | p. 54 |

Mixing Problems | p. 64 |

Exact Differential Equations | p. 73 |

Existence and Uniqueness of Solutions | p. 90 |

Dependence of Solutions on Initial Conditions | p. 102 |

Autonomous Equations and Stability | p. 107 |

The Daredevil Skydiver | p. 120 |

Modeling and Applications | p. 123 |

Modeling Population Growth | p. 124 |

Models and the Real World | p. 138 |

Personal Finance | p. 143 |

Electrical Circuits | p. 152 |

The Spruce Budworm | p. 158 |

Social Security, Now or Later | p. 161 |

Second-Order Equations | p. 163 |

Definitions and Examples | p. 163 |

Second-Order Equations and Systems | p. 174 |

Linear, Homogeneous Equations with Constant Coefficients | p. 179 |

Harmonic Motion | p. 190 |

Inhomogeneous Equations; the Method of Undetermined Coefficients | p. 199 |

Variation of Parameters | p. 209 |

Forced Harmonic Motion | p. 215 |

Nonlinear Oscillators | p. 228 |

The Laplace Transform | p. 231 |

The Definition of the Laplace Transform | p. 232 |

Basic Properties of the Laplace Transform | p. 241 |

The Inverse Laplace Transform | p. 248 |

Using the Laplace Transform to Solve Differential Equations | p. 256 |

Discontinuous Forcing Terms | p. 266 |

The Delta Function | p. 280 |

Convolutions | p. 287 |

Summary | p. 298 |

Forced Harmonic Oscillators | p. 299 |

Numerical Methods | p. 301 |

Euler's Method | p. 302 |

Runge-Kutta Methods | p. 314 |

Numerical Error Comparisons | p. 322 |

Practical Use of Solvers | p. 327 |

A Cautionary Tale | p. 332 |

Numerical Error Comparison | p. 334 |

Matrix Algebra | p. 335 |

Vectors and Matrices | p. 335 |

The Geometry of Systems of Linear Equations | p. 348 |

Solving Systems of Equations | p. 353 |

Properties of Solution Sets | p. 362 |

Subspaces | p. 371 |

Determinants | p. 384 |

An Introduction to Systems | p. 396 |

Definitions and Examples | p. 396 |

Geometric Interpretation of Solutions | p. 405 |

Qualitative Analysis | p. 417 |

Linear Systems | p. 425 |

Long-Term Behavior of Solutions | p. 441 |

Linear Systems with Constant Coefficients | p. 444 |

Overview of the Technique | p. 444 |

Planar Systems | p. 452 |

Phase Plane Portraits | p. 466 |

Higher Dimensional Systems | p. 484 |

The Exponential of a Matrix | p. 492 |

Qualitative Analysis of Linear Systems | p. 510 |

Higher-Order Linear Equations | p. 515 |

Inhomogeneous Linear Systems | p. 528 |

Phase Plane Portraits | p. 538 |

Oscillations of Linear Molecules | p. 539 |

Nonlinear Systems | p. 545 |

The Linearization of a Nonlinear System | p. 545 |

Long-Term Behavior of Solutions | p. 559 |

Invariant Sets and the Use of Nullclines | p. 566 |

Long-Term Behavior of Solutions to Planar Systems | p. 574 |

Conserved Quantities | p. 586 |

Nonlinear Mechanics | p. 592 |

The Method of Lyapunov | p. 609 |

Predator-Prey Systems | p. 619 |

Human Immune Response to Infectious Disease | p. 631 |

Analysis of Competing Species | p. 634 |

Series Solutions to Differential Equations | p. 637 |

Review of Power Series | p. 638 |

Series Solutions Near Ordinary Points | p. 650 |

Legendre's Equation | p. 662 |

Types of Singular Points--Euler's Equation | p. 668 |

Series Solutions Near Regular Singular Points | p. 677 |

Solutions in the Exceptional Cases | p. 690 |

Bessel's Equation and Bessel Functions | p. 701 |

Fourier Series | p. 712 |

Computation of Fourier Series | p. 713 |

Convergence of Fourier Series | p. 724 |

Fourier Cosine and Sine Series | p. 733 |

The Complex Form of a Fourier Series | p. 738 |

The Discrete Fourier Transform and the FFT | p. 741 |

Partial Differential Equations | p. 750 |

Derivation of the Heat Equation | p. 750 |

Separation of Variables for the Heat Equation | p. 756 |

The Wave Equation | p. 768 |

Laplace's Equation | p. 778 |

Laplace's Equation on a Disk | p. 785 |

Sturm Liouville Problems | p. 791 |

Orthogonality and Generalized Fourier Series | p. 801 |

Temperature in a Ball--Legendre Polynomials | p. 810 |

Time Dependent PDEs in Higher Dimension | p. 814 |

Domains with Circular Symmetry--Bessel Functions | p. 821 |

Answers to Odd-Numbered Problems | p. 1 |

Index | p. 1 |

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