A Course in Ordinary Differential Equations, 2nd Edition

## A Course in Ordinary Differential Equations, 2nd Edition

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A Course in Differential Equations with Boundary Value Problems, 2nd Edition by Wirkus, Swift, and Szypowski is now published!! Click on the book to the left to go to the webpage for it.

For A Course in Ordinary Differential Equations (book to the right), find us on the MATLAB website, Maple website, or Mathematica website.

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Summary

Author Biographies

Features

New to the Second Edition

Computer Labs

Typos and Errata

## Summary

A Course in Ordinary Differential Equations, Second Edition teaches students how to use analytical and numerical solution methods in typical engineering, physics, and mathematics applications. Lauded for its extensive computer code and student-friendly approach, the first edition of this popular textbook was the first on ordinary differential equations (ODEs) to include instructions on using MATLAB, Mathematica, and Maple. This second edition reflects the feedback of students and professors who used the first edition in the classroom.

Chapter 1: Traditional First-Order Differential Equations
1.1. Introduction to First-Order Equations
1.2. Separable Differential Equations
1.3. Linear Equations
1.4. Some Physical Models Arising as Separable Equations
1.5. Exact Equations
1.6. Special Integrating Factors and Substitution Methods

Chapter 2: Geometrical and Numerical Methods for First-Order Equations
2.1. Direction Fields---the Geometry of Differential Equations
2.2. Existence and Uniqueness for First-Order Equations
2.3. First-Order Autonomous Equations---Geometrical Insight
2.4. Modeling in Population Biology
2.5. Numerical Approximation: Euler and Runge-Kutta Methods
2.6. An Introduction to Autonomous Second-Order Equations

Chapter 3: Elements of Higher-Order Linear Equations
3.1. Introduction to Higher-Order Equations
3.2. Linear Independence and the Wronskian
3.3. Reduction of Order---the Case n = 2
3.4. Numerical Considerations for nth-Order Equations
3.5. Essential Topics from Complex Variables
3.6. Homogeneous Equations with Constant Coefficients
3.7. Mechanical and Electrical Vibrations

Chapter 4: Techniques of Nonhomogeneous Higher-Order Linear Equations
4.1. Nonhomogeneous Equations
4.2. Method of Undetermined Coefficients via Superposition
4.3. Method of Undetermined Coefficients via Annihilation
4.4. Exponential Response and Complex Replacement
4.5. Variation of Parameters
4.6. Cauchy-Euler (Equidimensional) Equation
4.7. Forced Vibrations

Chapter 5: Fundamentals of Systems of Differential Equations
5.1. Useful Terminology
5.2. Gaussian Elimination
5.3. Vector Spaces and Subspaces
5.4. Eigenvalues and Eigenvectors
5.5. A General Method, Part I: Solving Systems with Real & Distinct or Complex Eigenvalues
5.6. A General Method, Part II: Solving Systems with Repeated Real Eigenvalues
5.7. Matrix Exponentials
5.8. Solving Linear Nonhomogeneous Systems of Equations

Chapter 6: Geometrical Approaches and Applications of Systems of First-Order Equations
6.1. An Introduction to the Phase Plane
6.2. Nonlinear Equations and Phase Plane Analysis
6.3. Bifurcations
6.4. Epidemiological Models
6.5. Models in Ecology

Chapter 7: Laplace Transforms
7.1. Introduction
7.2. Fundamentals of the Laplace Transform
7.3. The Inverse Laplace Transform
7.4. Translated Functions, Delta Function, and Periodic Functions
7.5. The s-Domain and Poles
7.6. Solving Linear Systems using Laplace Transforms
7.7. The Convolution

Chapter 8: Series Methods
8.1. Power Series Representations of Functions
8.2. The Power Series Method
8.3. Ordinary and Singular Points
8.4. The Method of Frobenius
8.5. Bessel Functions

Appendix A: An Introduction to MATLAB, Maple, and Mathematica
A.1. MATLAB
A.2. Maple
A.3. Mathematica

Apprendix B: Selected Topics from Linear Algebra
B.1. A Primer on Matrix Algebra
B.2. Matrix Inverses, Cramer's Rule
B.3. Linear Transformations
B.4. Coordinates and Change of Basis

References
Index
A Review, Computer Labs, and Projects appear at the end of each chapter.

## Author Biographies

Stephen A. Wirkus completed his Ph.D. at Cornell University under the direction of Richard Rand. He began guiding undergraduate research projects while in graduate school and came to Cal Poly Pomona in 2000 after being a Visiting Professor at Cornell for a year. He co-founded the Applied Mathematical Sciences Summer Institute (AMSSI), an undergraduate research program jointly hosted by Loyola Marymount University, that ran from 2005 through 2007. He came to Arizona State University in 2007 as a tenured Associate Professor and won the 2013 Professor of the Year Award at ASU as well as the 2011 NSF AGEP Mentor of the Year award. He was a Visiting MLK Professor at the Massachusetts Institute of Technology in 2013-2014. He has guided over 80 undergraduate students in research and has served as Chair for 4 M.S. students, and 2 Ph.D. students. He has over 30 publications and technical reports with over 40 students and has received grants from the NSF and NSA for guiding undergraduate research.

Randall J. Swift completed his Ph.D. at the University of California, Riverside under the direction of M. M. Rao. He began his career at Western Kentucky University and taught there for nearly a decade before moving to Cal Poly Pomona in 2001 as a tenured Associate Professor. He is active in research and teaching, having authored more than 80 journal articles, three research monographs and three textbooks in addition to serving as Chair for 25 M.S. students. Now a Professor, he received the 2011-12 Ralph W. Ames Distinguished Research Award from the College of Science at Cal Poly Pomona. The award honors Swift for his outstanding research in both pure and applied mathematics, and his contributions to the mathematics field as a speaker, journal editor, and principal investigator on numerous grants. He was also a Visiting Professor in 2007-2008 at The Australian National University in Canberra Australia as well as having taught at the Claremont Colleges.

## Features

• Describes analytical and numerical methods for studying ODEs
• Shows students how to effectively use MATLAB, Maple, and Mathematica in practice, assuming no prior knowledge of the software packages
• Covers essential linear algebra topics, such as eigenvectors, bases, and transformations, to improve students' understanding of differential equations
• Includes numerous problems of varying levels of difficulty for applied and pure math majors as well as for engineers and other nonmathematicians
• Offers answers to most of the odd problems in the back of the book
• Contains reviews and projects at the end of each chapter
• Solutions manual available upon qualifying course adoption

## New to the Second Edition

• Moves the computer codes to Computer Labs at the end of each chapter, which gives professors flexibility in using the technology
• Covers linear systems in their entirety before addressing applications to nonlinear systems
• Includes new sections on complex variables, the exponential response formula for solving nonhomogeneous equations, forced vibrations, and nondimensionalization
• Highlights new applications and modeling in many fields
• Presents exercise sets that progress in difficulty
• Contains color graphs to help students better understand crucial concepts in ODEs
• Provides updated and expanded projects in each chapter
• Suitable for a first undergraduate course, the book includes all the basics necessary to prepare students for their future studies in mathematics, engineering, and the sciences. It presents the syntax from MATLAB, Maple, and Mathematica to give students a better grasp of the theory and gain more insight into real-world problems. Along with covering traditional topics, the text describes a number of modern topics, such as direction fields, phase lines, the Runge-Kutta method, and epidemiological and ecological models. It also explains concepts from linear algebra so that students acquire a thorough understanding of differential equations.

## Typos and Errata

Typos last updated on January 18, 2017, 1:59PM MST.

Please locate the Version Date of the book. To do so, turn to the page in the front with the copyright (immediately preceding the dedication page). In the middle of this page, you will see a Version Date, e.g., Version Date: 20141104
Go to the appropriate typo list below.

Typos from Version Date: 20141104

• p. 39, Prob 15, "Find the concentration of salt at time t."
• p. 77, Chapter 1 Mathematica Computer Code Example 4 is missing:
Quit[ ]
de[x_] = (x^2 + 1) y'[x] + 4 x y[x]
desol = NDSolve[{de[x] == x, y[0] == 10}, y, {x, 0, 4}]
p1 = Plot[Evaluate[y[x] /. %], {x, 0, 4}, PlotStyle -> Blue, AxesLabel -> {"x", "y"}]
sol1[x_] = (x^4/4 + x^2/2 + 10)/(x^2 + 1)^2
p2 = Plot[sol1[x], {x, 0, 4}, PlotStyle -> Black, AxesLabel -> {"x", "y"}]
Show[p1, p2]

• p. 86, Prob 24, point is (0,1).
• p. 138. Replace line beginning with steps with
steps=round((xf-x0)/h)
which just rounds the values in case it is not an integer.
• p. 144. In the line immediately preceding Maple Example 3, ignore everything before "display([eq2,eq3,eq4,eq5,eq6])

• p. 167, Ex. 4, "-1≤ x ≤ 1" in third line and last line of solution.
• p. 191, Prob 43, equation should be z''+z=0
• p. 191, Prob 44, equation should be z''+4z=0
• p. 191, Prob 45, "show that exp((a+ib)x) is a solution to z'=(a+ib)z."
• p. 201, Ex. 11, denominator outside square root should be "a_2".
• p. 202, Prob 3, equation should be y''-3y'-4y=0.
• p. 225. On the last line of the page, "g(x)" should be "g(r)"
• p. 265. The last two terms on the left hand side of the formula in the middle of page should be A_1 and A_2 (not k_1 and k_2).
• p. 277. The last formula line should be "(1/45)*[12sin(x)-24cos(2x)]."
• p. 301. Last equation before problems should have "[(sqrt(7)/2)*ln(x)]" as the argument for both sine and cosine in the two numerators.
• p. 351. Directions to Problems 47-50: "...find bases of the left nullspace and row space. Then repeat..."
• p. 365. The exponent in the middle term of each equation x(t), y(t), z(t) should be "-6t" (the negative is missing).
• p. 379. Eq. (5.95). The exponent of exp should have lambda=2.
• p. 379. Eq. (5.96). Carrying from above, "exp(2t)" need to multiply the matrix-vector expression (after first =) and the resulting vector (after the second =).
• p. 386. Eq. (5.110). The top right entry (-1/3)*exp(3) + (1/3)*exp(-3)
• p. 388. Eq. (5.115). The last column should be [6t*exp(2t)+exp(2t)-exp(-t), -9texp(2t)-2exp(-t)+2exp(2t), 3exp(2t)]
• p. 407. Last line before Example 5 has an extra paranthesis (remove the ")" at the end).
• p. 444. In the line immediately after Eq. (6.20), the middle equilibrium is "(-sqrt(r),0)"
• p. 469. Formula in Problem 12(d). The numerator on the right side should be "ν" (not "α")
• p. 557. In Problem 9, the second equation should be "x'-3x+y'-3y=2" (remove the ' from -3y')
• p. 503. The end of the 5th line should be "t=1"
• p. 513. Correction to Definition 7.2.1:   change "t>0" to include 0 also; that is, make it "t ≥ 0". Correct this in Examples 1-4 that follow.
• p. 517. The lower bound on t in Theorem 7.2.2 should be "0" (instead of "a").
• p. 667. 3rd line of code should refer to "x":
SinePlot=zeros(size(x))
• p. 702. Answer to 1.3.9: y=(1+x^2)(x-arctan(x))+(1+x^2)+C
• p. 702. Answer to 1.4.17: 11.45min
• p. 702. Answer to 1.4.19: 1.195 years
• p. 702. Answer to 1.4.21: y=+/- 2(a^2)/x
• p. 703. Answer to Chap 1 Review Prob 29: y = (x^4)/5+C/x
• p. 703. Answer to Chap 1 Review Prob 33: y^(-2)=exp(2x)*((1/2)*exp(-2x)(2x-1)+C)
• p. 748. Answer to 3.1.21: Q(D)P(D)(y)= -sin(x) + 3xcos(x); P(D)Q(D)(y)=3xcos(x) + 2sin(x)
• p. 748. Answer to 3.2.3: correct but for x ≠ 2
• p. 748. Answer to 3.2.17: Linearly dependent, e.g., C1=0, C2=exp(-4), C3=-1.
• p. 749. Answer to 3.2.29(iii): W(0)=7/2.
• p. 750. Answer to 3.4.11(i): Change 2nd IC to u_2(1)=0
• p. 750. Answer to 3.4.13(i): Change 3rd eq to u_3'=8u_1/x^3
• p. 750. Answer to 3.5.11: z=+/- 1
• p. 750. Answer to 3.5.13: change "i" to "-i".
• p. 750. Answer to 3.5.17: change 2nd part to p(2i)=-7-2i
• p. 750. Answer to 3.5.19: change 2nd part to p(1+i)=-i
• p. 750. Answer to 3.5.21: p(2i)=-exp(2i)/3; p(1-i)=1-i(1/5 + i2/5)
• p. 750. Answer to 3.5.29: change argument of arctan to arctan(1/2)
• p. 750. Answer to 3.5.35: +sin(x)+i*cos(x)+C
• p. 750. Answer to 3.5.43: z'=I*exp(I*x), z''=-exp(i*x) gives z''+z=0.
• p. 751. Answer to 3.7.3: 1st part is x(t)=(1/4)*sin(8t)
• p. 751. Answer to 3.7.25: k>1
• p. 751. Answer to 3.7.29: (1/2)>m
• p. 752. Answer to Chap 3 Review Prob 51: 2nd part is p(1+i)=(1-i)/6
• p. 752. Answer to Chap 3 Review Prob 53: p(3i)=-exp(3i)/5, p(i)=exp(i)/3

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