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A Course in Differential Equations with Boundary-Value Problems, 2nd Edition

A Course in Differential Equations with Boundary-Value Problems, 2nd Edition

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The webpage for A Course in Ordinary Differential Equations, 2nd Edition by Wirkus and Swift can be found by clicking on the book to the left.







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Summary

Table of Contents

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.
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Table of Contents

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

Chapter 9: Boundary-Value Problems and Fourier Series
  9.1. Two-Point Boundary-Value Problems
  9.2. Orthogonal Functions and Fourier Series
  9.3. Even, Odd, and Discontinuous Functions
  9.4. Simple Eigenvalue-Eigenfunction Problems
  9.5. Sturm-Liouville Theory
  9.6. Generalized Fourier Series

Chapter 10: Partial Differential Equations
  10.1. Separable Linear Partial Differential Equations
  10.2. Heat Equation
  10.3. Wave Equation
  10.4. Laplace Equation
  10.5. Nonhomogeneous Boundary Conditions
  10.6. Non-Cartesian Coordinate Systems

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

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

Ryan S. Szypowski completed his Ph.D. at the University of California, San Diego under the direction of Michael Holst. He was hired at Cal Poly Pomona in 2011 and became a tenured Associate Professor in 2016. His main research is in the area of computational methods for approximation of solutions to partial differential equations, but studies broad problems from applied mathematics and science with his students and colleagues. He had an active NSF research grant from 2012 to 2015 to study adaptive techniques in nite element exterior calculus and has won numerous smaller grants to support his work with students. In 2015, he won the Department of Mathematics and Statistics Excellence in Teaching Award.
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Features

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New to the Second Edition


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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 First Printing

  • p. 73, Prob 24, point is (0,1).
  • p. 321. Eq. (5.110). The top right entry (-1/3)*exp(3) + (1/3)*exp(-3)
  • p. 557. In Problem 9, the second equation should be "x'-3x+y'-3y=2" (remove the ' from -3y')




  • For First Edition Typos , click on the link.

    Thank you for reporting the above typos and errata. We appreciate your support!!



    Send additional ones to:
    Stephen Wirkus (e-mail: swirkus "at" asu "dot" edu)
    or
    Randall Swift (e-mail: rjswift "at" cpp "dot" edu)


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