Differential Equations With Mathematica

, by Abell, Martha L.; Braselton, James P.

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ISBN: 9780120415397 | 0120415399
Cover: Paperback
Copyright: 9/1/1993

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The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields, especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners. The book/CD-ROM package contains built-in commands that lets the user solve problems directly using graphical solutions.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* CD-ROM contains all Mathematica inputs from the text* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica

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
Table of Contents provided by Rittenhouse. All Rights Reserved.

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Differential Equations With Mathematica

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