Differential Equations and Boundary Value Problems: Computing and Modeling, 6th edition

Published by Pearson (February 3, 2022) © 2023

  • C Henry Edwards University of Georgia, Athens
  • David E. Penney University of Georgia, Athens
  • David Calvis Baldwin Wallace University

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For 1-semester sophomore- or junior-level Differential Equations courses.

Balances concepts, visualization and applications

Differential Equations and Boundary Value Problems fosters the conceptual development and geometric visualization essential to science and engineering students. Manual methods complement the computer-based methods that illuminate qualitative phenomena, opening up a wider range of more realistic applications.

One text now meets all course needs. Courses not covering boundary value problems can use the 6th Edition with no added cost for that material. This revision also adds and updates content throughout, including an expanded Application Module that discusses COVID-19.

Hallmark features of this title

  • Emphasis on numerical methods includes early introduction of numerical solution techniques, mathematical modeling, stability and qualitative properties of differential equations, with generic numerical algorithms that can be implemented in various technologies.
  • Application Modules follow key sections, most with computing projects that reinforce the corresponding text sections.
  • Approximately 2000 problems range from computational to applied and conceptual problems. 
  • An expansive answer section includes answers to most odd- and even-numbered problems. 
  • Emphasis on technology and ODEs explores newer methods of computing differential equations, covering the software systems tailored specifically to differential equations as well as Maple, Mathematica and MATLAB.

New and updated features of this title

  • New content includes a new application of differential equations to the life sciences in Application Module 6.4: The Rayleigh, van der Pol, and FitzHugh-Nagumo Equations; The SIR Model and COVID-19. Characterized by the same careful and thorough exposition found throughout the text, this new unit gives students yet another perspective about differential equations.
  • Extensively revised design:
    • New use of full color enhances graphs and figures so that students can more easily discern different solutions in the figures.
    • Added marginal notes aid in understanding the mathematics in the text; easier identification of application topics in the exercise set includes new run-in problem titles; new Your Turn headers in the Application Modules now clarify where the exposition ends and the students' investigations begin.
  • 16 new Interactive Figures illustrate how interactive computer applications with slider bars or touchpad controls can be used to change initial values or parameters in a differential equation, allowing students to immediately see in real time the resulting changes in the structure of its solutions.
    • Using a mouse or touchpad, the initial point for an initial value problem can be dragged to a new location, and the corresponding solution curve is automatically redrawn and dragged along with its initial point. For examples, see Figures 1.3.5 and 3.2.4.

Features of MyLab Math for the 6th Edition

  • Additional Exercises with immediate feedback: Over 1000 assignable exercises are based on the textbook exercises, and regenerate algorithmically to give students unlimited opportunity for practice and mastery. MyLab Math provides helpful feedback when students enter incorrect answers and includes optional learning aids including Help Me Solve This, View an Example, videos, and an eText.
  • New Set-up & Solve Exercises require students to describe how they will set up and approach the problem. This reinforces conceptual understanding of the process applied in approaching the problem, promotes long-term retention of the skill, and mirrors what students will be expected to do on a test.
  • Instructional videos provide meaningful support as a learning aid within exercises, alongside key examples in the eText, or for self-study within the Video & Resource Library. Instructors can assign videos within MyLab homework, use videos in class, or offer as a supplementary resource on specific topics.
  • Early Alerts are now included with Performance Analytics and use predictive analytics to identify struggling students, even if their assignment scores are not a cause for concern. In both Performance Analytics and Early Alerts, instructors can email students individually or by group to provide feedback. 

1. First-Order Differential Equations

  • 1.1 Differential Equations and Mathematical Models
  • 1.2 Integrals as General and Particular Solutions
  • 1.3 Slope Fields and Solution Curves
  • 1.4 Separable Equations and Applications
  • 1.5 Linear First-Order Equations
  • 1.6 Substitution Methods and Exact Equations

2. Mathematical Models and Numerical Methods

  • 2.1 Population Models
  • 2.2 Equilibrium Solutions and Stability
  • 2.3 Acceleration - Velocity Models
  • 2.4 Numerical Approximation: Euler's Method
  • 2.5 A Closer Look at the Euler Method
  • 2.6 The Runge - Kutta Method

3. Linear Equations of Higher Order

  • 3.1 Introduction: Second-Order Linear Equations
  • 3.2 General Solutions of Linear Equations
  • 3.3 Homogeneous Equations with Constant Coefficients
  • 3.4 Mechanical Vibrations
  • 3.5 Nonhomogeneous Equations and Undetermined Coefficients
  • 3.6 Forced Oscillations and Resonance
  • 3.7 Electrical Circuits
  • 3.8 Endpoint Problems and Eigenvalues

4. Introduction to Systems of Differential Equations

  • 4.1 First-Order Systems and Applications
  • 4.2 The Method of Elimination
  • 4.3 Numerical Methods for Systems

5. Linear Systems of Differential Equations

  • 5.1 Matrices and Linear Systems
  • 5.2 The Eigenvalue Method for Homogeneous Systems
  • 5.3 A Gallery of Solution Curves of Linear Systems
  • 5.4 Second-Order Systems and Mechanical Applications
  • 5.5 Multiple Eigenvalue Solutions
  • 5.6 Matrix Exponentials and Linear Systems
  • 5.7 Nonhomogeneous Linear Systems

6. Nonlinear Systems and Phenomena

  • 6.1 Stability and the Phase Plane
  • 6.2 Linear and Almost Linear Systems
  • 6.3 Ecological Models: Predators and Competitors
  • 6.4 Nonlinear Mechanical Systems
  • 6.5 Chaos in Dynamical Systems

7. Laplace Transform Methods

  • 7.1 Laplace Transforms and Inverse Transforms
  • 7.2 Transformation of Initial Value Problems
  • 7.3 Translation and Partial Fractions
  • 7.4 Derivatives, Integrals, and Products of Transforms
  • 7.5 Periodic and Piecewise Continuous Input Functions
  • 7.6 Impulses and Delta Functions

8. Power Series Methods

  • 8.1 Introduction and Review of Power Series
  • 8.2 Series Solutions Near Ordinary Points
  • 8.3 Regular Singular Points
  • 8.4 Method of Frobenius: The Exceptional Cases
  • 8.5 Bessel's Equation
  • 8.6 Applications of Bessel Functions

9. Fourier Series Methods and Partial Differential Equations

  • 9.1 Periodic Functions and Trigonometric Series
  • 9.2 General Fourier Series and Convergence
  • 9.3 Fourier Sine and Cosine Series
  • 9.4 Applications of Fourier Series
  • 9.5 Heat Conduction and Separation of Variables
  • 9.6 Vibrating Strings and the One-Dimensional Wave Equation
  • 9.7 Steady-State Temperature and Laplace's Equation

10. Eigenvalue Methods and Boundary Value Problems

  • 10.1 Sturm - Liouville Problems and Eigenfunction Expansions
  • 10.2 Applications of Eigenfunction Series
  • 10.3 Steady Periodic Solutions and Natural Frequencies
  • 10.4 Cylindrical Coordinate Problems
  • 10.5 Higher-Dimensional Phenomena 
References for Further Study
Appendix: Existence and Uniqueness of Solutions
Answers to Selected Problems
Index

About our authors

Henry Edwards  is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960; he retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence.

His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students. In 2013 Prof. Edwards was named a Fellow of the American Mathematical Society.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 under the direction of Prof. L. Bruce Treybig while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms.

Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous university-wide teaching awards and directed several doctoral dissertations and 7 undergraduate research projects. He authored research papers in number theory and topology, and was the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra and liberal arts mathematics. Penney passed away in 2014.

David T. Calvis is Professor of Mathematics at Baldwin Wallace University near Cleveland, Ohio.  He completed a Ph.D. in complex analysis from the University of Michigan in 1988 under the direction of Fred Gehring.  While at Michigan he also received a Master's degree in Computer, Information, and Control Engineering.  Having initially served at Hillsdale College in Michigan, he has been at Baldwin Wallace since 1990, most recently assisting with the creation of an Applied Mathematics program there.  He has received a number of teaching awards, including BWU's Strosacker Award for Excellence in Teaching and Student Senate Teaching Award.  He is the author of materials dealing with the use of computer algebra systems in mathematics instruction, and has extensive classroom experience teaching differential equations and related topics.

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