Preface | p. xiii |
Introduction to Differential Equations | p. 1 |
Definitions and Concepts | p. 2 |
Solutions of Differential Equations | p. 6 |
Initial and Boundary-Value Problems | p. 18 |
Direction Fields | p. 26 |
First-Order Ordinary Differential Equations | p. 41 |
Theory of First-Order Equations: A Brief Discussion | p. 41 |
Separation of Variables | p. 46 |
Application: Kidney Dialysis | p. 55 |
Homogeneous Equations | p. 59 |
Application: Models of Pursuit | p. 64 |
Exact Equations | p. 69 |
Linear Equations | p. 74 |
Integrating Factor Approach | p. 75 |
Variation of Parameters and the Method of Undetermined Coefficients | p. 86 |
Application: Antibiotic Production | p. 89 |
Numerical Approximations of Solutions to First-Order Equations | p. 92 |
Built-In Methods | p. 92 |
Application: Modeling the Spread of a Disease | p. 97 |
Other Numerical Methods | p. 103 |
Applications of First-Order Ordinary Differential Equations | p. 119 |
Orthogonal Trajectories | p. 119 |
Application: Oblique Trajectories | p. 129 |
Population Growth and Decay | p. 132 |
The Malthus Model | p. 132 |
The Logistic Equation | p. 138 |
Application: Harvesting | p. 148 |
Application: The Logistic Difference Equation | p. 152 |
Newton's Law of Cooling | p. 157 |
Free-Falling Bodies | p. 163 |
Higher-Order Differential Equations | p. 175 |
Preliminary Definitions and Notation | p. 175 |
Introduction | p. 175 |
The nth-Order Ordinary Linear Differential Equation | p. 180 |
Fundamental Set of Solutions | p. 185 |
Existence of a Fundamental Set of Solutions | p. 191 |
Reduction of Order | p. 193 |
Solving Homogeneous Equations with Constant Coefficients | p. 196 |
Second-Order Equations | p. 196 |
Higher-Order Equations | p. 200 |
Application: Testing for Diabetes | p. 211 |
Introduction to Solving Nonhomogeneous Equations with Constant Coefficients | p. 216 |
Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients | p. 222 |
Second-Order Equations | p. 223 |
Higher-Order Equations | p. 239 |
Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters | p. 248 |
Second-Order Equations | p. 248 |
Higher-Order Nonhomogeneous Equations | p. 252 |
Cauchy-Euler Equations | p. 255 |
Second-Order Cauchy-Euler Equations | p. 255 |
Higher-Order Cauchy-Euler Equations | p. 261 |
Variation of Parameters | p. 265 |
Series Solutions | p. 268 |
Power Series Solutions about Ordinary Points | p. 268 |
Series Solutions about Regular Singular Points | p. 281 |
Method of Frobenius | p. 283 |
Application: Zeros of the Bessel Functions of the First Kind | p. 295 |
Application: The Wave Equation on a Circular Plate | p. 298 |
Nonlinear Equations | p. 304 |
Applications of Higher-Order Differential Equations | p. 321 |
Harmonic Motion | p. 321 |
Simple Harmonic Motion | p. 321 |
Damped Motion | p. 332 |
Forced Motion | p. 346 |
Soft Springs | p. 365 |
Hard Springs | p. 368 |
Aging Springs | p. 370 |
Application: Hearing Beats and Resonance | p. 372 |
The Pendulum Problem | p. 373 |
Other Applications | p. 387 |
L-R-C Circuits | p. 387 |
Deflection of a Beam | p. 390 |
Bode Plots | p. 393 |
The Catenary | p. 398 |
Systems of Ordinary Differential Equations | p. 411 |
Review of Matrix Algebra and Calculus | p. 411 |
Defining Nested Lists, Matrices, and Vectors | p. 411 |
Extracting Elements of Matrices | p. 416 |
Basic Computations with Matrices | p. 419 |
Eigenvalues and Eigenvectors | p. 422 |
Matrix Calculus | p. 426 |
Systems of Equations: Preliminary Definitions and Theory | p. 427 |
Preliminary Theory | p. 429 |
Linear Systems | p. 446 |
Homogeneous Linear Systems with Constant Coefficients | p. 454 |
Distinct Real Eigenvalues | p. 454 |
Complex Conjugate Eigenvalues | p. 461 |
Alternate Method for Solving Initial-Value Problems | p. 474 |
Repeated Eigenvalues | p. 477 |
Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential | p. 485 |
Undetermined Coefficients | p. 485 |
Variation of Parameters | p. 490 |
The Matrix Exponential | p. 498 |
Numerical Methods | p. 506 |
Built-In Methods | p. 506 |
Application: Controlling the Spread of a Disease | p. 513 |
Euler's Method | p. 525 |
Runge-Kutta Method | p. 531 |
Nonlinear Systems, Linearization, and Classification of Equilibrium Points | p. 535 |
Real Distinct Eigenvalues | p. 535 |
Repeated Eigenvalues | p. 543 |
Complex Conjugate Eigenvalues | p. 548 |
Nonlinear Systems | p. 552 |
Applications of Systems of Ordinary Differential Equations | p. 567 |
Mechanical and Electrical Problems with First-Order Linear Systems | p. 567 |
L-R-C Circuits with Loops | p. 567 |
L-R-C Circuit with One Loop | p. 568 |
L-R-C Circuit with Two Loops | p. 571 |
Spring-Mass Systems | p. 574 |
Diffusion and Population Problems with First-Order Linear Systems | p. 576 |
Diffusion through a Membrane | p. 576 |
Diffusion through a Double-Walled Membrane | p. 578 |
Population Problems | p. 583 |
Applications that Lead to Nonlinear Systems | p. 587 |
Biological Systems: Predator-Prey Interactions, The Lotka-Volterra System, and Food Chains in the Chemostat | p. 587 |
Physical Systems: Variable Damping | p. 604 |
Differential Geometry: Curvature | p. 611 |
Laplace Transform Methods | p. 617 |
The Laplace Transform | p. 618 |
Definition of the Laplace Transform | p. 618 |
Exponential Order | p. 621 |
Properties of the Laplace Transform | p. 623 |
The Inverse Laplace Transform | p. 629 |
Definition of the Inverse Laplace Transform | p. 629 |
Laplace Transform of an Integral | p. 635 |
Solving Initial-Value Problems with the Laplace Transform | p. 637 |
Laplace Transforms of Step and Periodic Functions | p. 645 |
Piecewise-Defined Functions: The Unit Step Function | p. 645 |
Solving Initial-Value Problems | p. 649 |
Periodic Functions | p. 652 |
Impulse Functions: The Delta Function | p. 661 |
The Convolution Theorem | p. 667 |
The Convolution Theorem | p. 667 |
Integral and Integrodifferential Equations | p. 669 |
Applications of Laplace Transforms, Part I | p. 672 |
Spring-Mass Systems Revisited | p. 672 |
L-R-C Circuits Revisited | p. 679 |
Population Problems Revisited | p. 687 |
Application: The Tautochrone | p. 689 |
Laplace Transform Methods for Systems | p. 691 |
Applications of Laplace Transforms, Part II | p. 708 |
Coupled Spring-Mass Systems | p. 708 |
The Double Pendulum | p. 714 |
Application: Free Vibration of a Three-Story Building | p. 720 |
Eigenvalue Problems and Fourier Series | p. 727 |
Boundary-Value Problems, Eigenvalue Problems, Sturm-Liouville Problems | p. 727 |
Boundary-Value Problems | p. 727 |
Eigenvalue Problems | p. 730 |
Sturm-Liouville Problems | p. 735 |
Fourier Sine Series and Cosine Series | p. 737 |
Fourier Sine Series | p. 737 |
Fourier Cosine Series | p. 746 |
Fourier Series | p. 749 |
Fourier Series | p. 749 |
Even, Odd, and Periodic Extensions | p. 758 |
Differentiation and Integration of Fourier Series | p. 764 |
Parseval's Equality | p. 768 |
Generalized Fourier Series | p. 770 |
Partial Differential Equations | p. 783 |
Introduction to Partial Differential Equations and Separation of Variables | p. 783 |
Introduction | p. 783 |
Separation of Variables | p. 785 |
The One-Dimensional Heat Equation | p. 787 |
The Heat Equation with Homogeneous Boundary Conditions | p. 787 |
Nonhomogeneous Boundary Conditions | p. 791 |
Insulated Boundary | p. 795 |
The One-Dimensional Wave Equation | p. 799 |
The Wave Equation | p. 799 |
D'Alembert's Solution | p. 806 |
Problems in Two Dimensions: Laplace's Equation | p. 810 |
Laplace's Equation | p. 810 |
Two-Dimensional Problems in a Circular Region | p. 817 |
Laplace's Equation in a Circular Region | p. 817 |
The Wave Equation in a Circular Region | p. 821 |
Other Partial Differential Equations | p. 836 |
Getting Started | p. 841 |
Introduction to Mathematica | p. 841 |
A Note Regarding Different Versions of Mathematica | p. 843 |
Getting Started with Mathematica | p. 843 |
Five Basic Rules of Mathematica Syntax | p. 849 |
Loading Packages | p. 850 |
A Word of Caution | p. 853 |
Getting Help from Mathematica | p. 854 |
Mathematica Help | p. 858 |
The Mathematica Menu | p. 863 |
Bibliography | p. 865 |
Index | p. 867 |
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