Differential Equations with Mathematica

by Martha L. Abell, James P. Braselton

Overview

Differential Equations with Mathematica, Fifth Edition uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists. Mathematica's diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica's built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, Mathematica can be used to perform the calculations encountered when solving a differential equation. Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica's outstanding graphics capabilities. - Demonstrates how to take advantage of the advanced features of Mathematica - Introduces the fundamental theory of ordinary and partial differential equations using Mathematica to solve typical problems of interest to students, instructors, scientists, and practitioners in many fields - Showcases practical applications and case studies drawn from biology, physics, and engineering

Product Details

ISBN-13:	9780323984362
Publisher:	Elsevier Science & Technology Books
Publication date:	01/18/2022
Sold by:	Barnes & Noble
Format:	eBook
Pages:	608
File size:	99 MB
Note:	This product may take a few minutes to download.

About the Author

Martha L. Abell and James P. Braselton are graduates of the Georgia Institute of Technology and the Ohio State University, respectively, and teach at Georgia Southern University, Statesboro where they have extensive experience instructing students at both the undergraduate and graduate levels. Other books by the authors include Differential Equations with Mathematica and Mathematica by Example.Martha L. Abell and James P. Braselton are graduates of the Georgia Institute of Technology and the Ohio State University, respectively, and teach at Georgia Southern University, Statesboro where they have extensive experience instructing students at both the undergraduate and graduate levels. Other books by the authors include Differential Equations with Mathematica and Mathematica by Example.

Table of Contents

	Preface	xiii
1	Introduction to Differential Equations	1
1.1	Definitions and Concepts	2
1.2	Solutions of Differential Equations	6
1.3	Initial and Boundary-Value Problems	18
1.4	Direction Fields	26
2	First-Order Ordinary Differential Equations	41
2.1	Theory of First-Order Equations: A Brief Discussion	41
2.2	Separation of Variables	46
	Application: Kidney Dialysis	55
2.3	Homogeneous Equations	59
	Application: Models of Pursuit	64
2.4	Exact Equations	69
2.5	Linear Equations	74
2.5.1	Integrating Factor Approach	75
2.5.2	Variation of Parameters and the Method of Undetermined Coefficients	86
	Application: Antibiotic Production	89
2.6	Numerical Approximations of Solutions to First-Order Equations	92
2.6.1	Built-In Methods	92
	Application: Modeling the Spread of a Disease	97
2.6.2	Other Numerical Methods	103
3	Applications of First-Order Ordinary Differential Equations	119
3.1	Orthogonal Trajectories	119
	Application: Oblique Trajectories	129
3.2	Population Growth and Decay	132
3.2.1	The Malthus Model	132
3.2.2	The Logistic Equation	138
	Application: Harvesting	148
	Application: The Logistic Difference Equation	152
3.3	Newton's Law of Cooling	157
3.4	Free-Falling Bodies	163
4	Higher-Order Differential Equations	175
4.1	Preliminary Definitions and Notation	175
4.1.1	Introduction	175
4.1.2	The nth-Order Ordinary Linear Differential Equation	180
4.1.3	Fundamental Set of Solutions	185
4.1.4	Existence of a Fundamental Set of Solutions	191
4.1.5	Reduction of Order	193
4.2	Solving Homogeneous Equations with Constant Coefficients	196
4.2.1	Second-Order Equations	196
4.2.2	Higher-Order Equations	200
	Application: Testing for Diabetes	211
4.3	Introduction to Solving Nonhomogeneous Equations with Constant Coefficients	216
4.4	Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients	222
4.4.1	Second-Order Equations	223
4.4.2	Higher-Order Equations	239
4.5	Nonhomogeneous Equations with Constant Coefficients: Variation of Parameters	248
4.5.1	Second-Order Equations	248
4.5.2	Higher-Order Nonhomogeneous Equations	252
4.6	Cauchy-Euler Equations	255
4.6.1	Second-Order Cauchy-Euler Equations	255
4.6.2	Higher-Order Cauchy-Euler Equations	261
4.6.3	Variation of Parameters	265
4.7	Series Solutions	268
4.7.1	Power Series Solutions about Ordinary Points	268
4.7.2	Series Solutions about Regular Singular Points	281
4.7.3	Method of Frobenius	283
	Application: Zeros of the Bessel Functions of the First Kind	295
	Application: The Wave Equation on a Circular Plate	298
4.8	Nonlinear Equations	304
5	Applications of Higher-Order Differential Equations	321
5.1	Harmonic Motion	321
5.1.1	Simple Harmonic Motion	321
5.1.2	Damped Motion	332
5.1.3	Forced Motion	346
5.1.4	Soft Springs	365
5.1.5	Hard Springs	368
5.1.6	Aging Springs	370
	Application: Hearing Beats and Resonance	372
5.2	The Pendulum Problem	373
5.3	Other Applications	387
5.3.1	L-R-C Circuits	387
5.3.2	Deflection of a Beam	390
5.3.3	Bode Plots	393
5.3.4	The Catenary	398
6	Systems of Ordinary Differential Equations	411
6.1	Review of Matrix Algebra and Calculus	411
6.1.1	Defining Nested Lists, Matrices, and Vectors	411
6.1.2	Extracting Elements of Matrices	416
6.1.3	Basic Computations with Matrices	419
6.1.4	Eigenvalues and Eigenvectors	422
6.1.5	Matrix Calculus	426
6.2	Systems of Equations: Preliminary Definitions and Theory	427
6.2.1	Preliminary Theory	429
6.2.2	Linear Systems	446
6.3	Homogeneous Linear Systems with Constant Coefficients	454
6.3.1	Distinct Real Eigenvalues	454
6.3.2	Complex Conjugate Eigenvalues	461
6.3.3	Alternate Method for Solving Initial-Value Problems	474
6.3.4	Repeated Eigenvalues	477
6.4	Nonhomogeneous First-Order Systems: Undetermined Coefficients, Variation of Parameters, and the Matrix Exponential	485
6.4.1	Undetermined Coefficients	485
6.4.2	Variation of Parameters	490
6.4.3	The Matrix Exponential	498
6.5	Numerical Methods	506
6.5.1	Built-In Methods	506
	Application: Controlling the Spread of a Disease	513
6.5.2	Euler's Method	525
6.5.3	Runge-Kutta Method	531
6.6	Nonlinear Systems, Linearization, and Classification of Equilibrium Points	535
6.6.1	Real Distinct Eigenvalues	535
6.6.2	Repeated Eigenvalues	543
6.6.3	Complex Conjugate Eigenvalues	548
6.6.4	Nonlinear Systems	552
7	Applications of Systems of Ordinary Differential Equations	567
7.1	Mechanical and Electrical Problems with First-Order Linear Systems	567
7.1.1	L-R-C Circuits with Loops	567
7.1.2	L-R-C Circuit with One Loop	568
7.1.3	L-R-C Circuit with Two Loops	571
7.1.4	Spring-Mass Systems	574
7.2	Diffusion and Population Problems with First-Order Linear Systems	576
7.2.1	Diffusion through a Membrane	576
7.2.2	Diffusion through a Double-Walled Membrane	578
7.2.3	Population Problems	583
7.3	Applications that Lead to Nonlinear Systems	587
7.3.1	Biological Systems: Predator-Prey Interactions, The Lotka-Volterra System, and Food Chains in the Chemostat	587
7.3.2	Physical Systems: Variable Damping	604
7.3.3	Differential Geometry: Curvature	611
8	Laplace Transform Methods	617
8.1	The Laplace Transform	618
8.1.1	Definition of the Laplace Transform	618
8.1.2	Exponential Order	621
8.1.3	Properties of the Laplace Transform	623
8.2	The Inverse Laplace Transform	629
8.2.1	Definition of the Inverse Laplace Transform	629
8.2.2	Laplace Transform of an Integral	635
8.3	Solving Initial-Value Problems with the Laplace Transform	637
8.4	Laplace Transforms of Step and Periodic Functions	645
8.4.1	Piecewise-Defined Functions: The Unit Step Function	645
8.4.2	Solving Initial-Value Problems	649
8.4.3	Periodic Functions	652
8.4.4	Impulse Functions: The Delta Function	661
8.5	The Convolution Theorem	667
8.5.1	The Convolution Theorem	667
8.5.2	Integral and Integrodifferential Equations	669
8.6	Applications of Laplace Transforms, Part I	672
8.6.1	Spring-Mass Systems Revisited	672
8.6.2	L-R-C Circuits Revisited	679
8.6.3	Population Problems Revisited	687
	Application: The Tautochrone	689
8.7	Laplace Transform Methods for Systems	691
8.8	Applications of Laplace Transforms, Part II	708
8.8.1	Coupled Spring-Mass Systems	708
8.8.2	The Double Pendulum	714
	Application: Free Vibration of a Three-Story Building	720
9	Eigenvalue Problems and Fourier Series	727
9.1	Boundary-Value Problems, Eigenvalue Problems, Sturm-Liouville Problems	727
9.1.1	Boundary-Value Problems	727
9.1.2	Eigenvalue Problems	730
9.1.3	Sturm-Liouville Problems	735
9.2	Fourier Sine Series and Cosine Series	737
9.2.1	Fourier Sine Series	737
9.2.2	Fourier Cosine Series	746
9.3	Fourier Series	749
9.3.1	Fourier Series	749
9.3.2	Even, Odd, and Periodic Extensions	758
9.3.3	Differentiation and Integration of Fourier Series	764
9.3.4	Parseval's Equality	768
9.4	Generalized Fourier Series	770
10	Partial Differential Equations	783
10.1	Introduction to Partial Differential Equations and Separation of Variables	783
10.1.1	Introduction	783
10.1.2	Separation of Variables	785
10.2	The One-Dimensional Heat Equation	787
10.2.1	The Heat Equation with Homogeneous Boundary Conditions	787
10.2.2	Nonhomogeneous Boundary Conditions	791
10.2.3	Insulated Boundary	795
10.3	The One-Dimensional Wave Equation	799
10.3.1	The Wave Equation	799
10.3.2	D'Alembert's Solution	806
10.4	Problems in Two Dimensions: Laplace's Equation	810
10.4.1	Laplace's Equation	810
10.5	Two-Dimensional Problems in a Circular Region	817
10.5.1	Laplace's Equation in a Circular Region	817
10.5.2	The Wave Equation in a Circular Region	821
10.5.3	Other Partial Differential Equations	836
Appendix	Getting Started	841
	Introduction to Mathematica	841
	A Note Regarding Different Versions of Mathematica	843
	Getting Started with Mathematica	843
	Five Basic Rules of Mathematica Syntax	849
	Loading Packages	850
	A Word of Caution	853
	Getting Help from Mathematica	854
	Mathematica Help	858
	The Mathematica Menu	863
	Bibliography	865
	Index	867

What People are Saying About This

From the Publisher

Updated resource focusing particularly on applications and Mathematica’s latest graphical capabilities

Editorial Reviews

Most topics covered include typical examples solved by both traditional methods, and by the computer math aid, Mathematica. The authors also use Mathematica to establish well-known algorithms for solving elementary differential equations. Practical problems for scientists and engineers are discussed, along with theoretical concepts. Many illustrations show what to type into Mathematica and just what comes out. Contains a glossary of commands. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

Stephen McDowall of Western Washington University says:
" The topics covered are extensive. The order and organization are good, especially in terms of increasing sophistication of the mathematics involved and in the complexity of the Mathematica programming necessary"
March 2003
Mark Lusk of The Colorado School of Mines says:
" I am considering the adoption of this book for my graduate class in Simulation and Modeling. I
make very heavy use of Mathematica and have a weekly computer lab. I am tempted to make this the sole, required text for that course"
March 2003

From the Publisher