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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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