Higher Education

Differential Equations, 4th Edition

  • includes DE Tools Printed Access Card
  • Paul Blanchard Boston University
  • Robert L. Devaney Boston University
  • Glen R. Hall Boston University
  • ISBN-10: 1133109039  |  ISBN-13: 9781133109037
  • 864 Pages
  • Previous Editions: 2006, 2002, 1998
  • © 2011 | Published
  • College Bookstore Wholesale Price = $140.75
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About

Overview

Incorporating an innovative modeling approach, this text for a one-semester differential equations course emphasizes conceptual understanding to help students relate information taught in the classroom to real-world experiences. Going beyond a traditional emphasis on technique, the authors focus on understanding how differential equations are formulated and interpreting their meaning to applied models from a variety of disciplines. A three-pronged qualitative, numeric, and analytic approach stresses visualizing differential equations geometrically, utilizing the latest computational technology to investigate the behavior of solutions, and predicting the behavior of solutions as they apply to models. The presentation weaves various points of view together so students become adept at moving between different representations to solve nonlinear differential equations equally well as traditional linear equations. Certain models reappear throughout the text as running themes to synthesize different concepts from multiple angles, and a dynamical systems focus emphasizes predicting the long-term behavior of these recurring models. Students will discover how to identify and harness the mathematics they will use in their careers, and apply it effectively outside the classroom.

Features and Benefits

  • The book's modeling approach emphasizes understanding of the meaning of variables and parameters in a differential equation and interpreting this meaning in an applied setting.
  • Revisited models in the text reinforce recurring themes and synthesize multiple concepts to solve practical problems. This helps students develop a natural intuition for how to approach an applied problem in real life without a road map.
  • A large number of well-chosen exercises integrate the qualitative, numerical, and analytic arguments behind a solution and require students to demonstrate understanding of the concepts behind the problem.
  • Lab activities at the end of each chapter offer deeper explorations of models, and require students to summarize their results in a report that demonstrates the necessary qualitative, numerical, and analytic arguments.
  • Certain exercises make optional use of specific applets in the DE Tools suite to harness the power of technology in solving differential equations.

Table of Contents

1. FIRST-ORDER DIFFERENTIAL EQUATIONS.
Modeling via Differential Equations. Analytic Technique: Separation of Variables. Qualitative Technique: Slope Fields. Numerical Technique: Euler's Method. Existence and Uniqueness of Solutions. Equilibria and the Phase Line. Bifurcations. Linear Equations. Integrating Factors for Linear Equations.
2. FIRST-ORDER SYSTEMS.
Modeling via Systems. The Geometry of Systems. Analytic Methods for Special Systems. Euler's Method for Systems. The Lorenz Equations.
3. LINEAR SYSTEMS.
Properties of Linear Systems and the Linearity Principle. Straight-Line Solutions. Phase Planes for Linear Systems with Real Eigenvalues. Complex Eigenvalues. Special Cases: Repeated and Zero Eigenvalues. Second-Order Linear Equations. The Trace-Determinant Plane. Linear Systems in Three Dimensions.
4. FORCING AND RESONANCE.
Forced Harmonic Oscillators. Sinusoidal Forcing. Undamped Forcing and Resonance. Amplitude and Phase of the Steady State. The Tacoma Narrows Bridge.
5. NONLINEAR SYSTEMS.
Equilibrium Point Analysis. Qualitative Analysis. Hamiltonian Systems. Dissipative Systems. Nonlinear Systems in Three Dimensions. Periodic Forcing of Nonlinear Systems and Chaos.
6. LAPLACE TRANSFORMS.
Laplace Transforms. Discontinuous Functions. Second-Order Equations. Delta Functions and Impulse Forcing. Convolutions. The Qualitative Theory of Laplace Transforms.
7. NUMERICAL METHODS.
Numerical Error in Euler's Method. Improving Euler's Method. The Runge-Kutta Method. The Effects of Finite Arithmetic.
8. DISCRETE DYNAMICAL SYSTEMS.
The Discrete Logistic Equation. Fixed Points and Periodic Points. Bifurcations. Chaos. Chaos in the Lorenz System.
APPENDICES.
A. Changing Variables.
B. The Ultimate Guess.
C. Complex Numbers and Euler's Formula.

What's New

  • New in-depth models on timely and relevant applications such as SIR infectious disease modeling.
  • Chapters on First-Order Systems, Linear Systems, and Forcing and Resonance have been extensively rewritten for better clarity and readability.
  • Several sections, particularly in Chapters 2-4, have been reorganized for teaching flexibility and to better fit into class time constraints.
  • Exercise sets have been thoroughly refreshed with updated problems throughout the text.

Supplements

All supplements have been updated in coordination with the main title. Select the main title's "About" tab, then select "What's New" for updates specific to title's edition.

For more information about these supplements, or to obtain them, contact your Learning Consultant.

Instructor Supplements

Solution Builder  (ISBN-10: 0495826715 | ISBN-13: 9780495826712)

Containing complete worked-out solutions to all exercises in the text this flexible, personalized online tool lets you easily build and save solution sets matched directly to your own homework assignments. Adopting instructors can sign up for an account by going to www.cengage.com/solutionbuilder.

Student Solutions Manual  (ISBN-10: 0495826723 | ISBN-13: 9780495826729)

Contains fully worked-out solutions to all of the odd-numbered exercises in the text.

List Price = $114.95  | CengageBrain Price = $114.95  | College Bookstore Wholesale Price = $86.50

DE Tools Instant Access Code  (ISBN-10: 0495562009 | ISBN-13: 9780495562009)

List Price = $65.00  | CengageBrain Price = $65.00  | College Bookstore Wholesale Price = $65.00

Student Supplements

Student Solutions Manual  (ISBN-10: 0495826723 | ISBN-13: 9780495826729)

Contains fully worked-out solutions to all of the odd-numbered exercises in the text.

List Price = $114.95  | CengageBrain Price = $114.95  | College Bookstore Wholesale Price = $86.50

DE Tools Instant Access Code  (ISBN-10: 0495562009 | ISBN-13: 9780495562009)

List Price = $65.00  | CengageBrain Price = $65.00  | College Bookstore Wholesale Price = $65.00

Meet the Author

Author Bio

Paul Blanchard

Paul Blanchard is Associate Professor of Mathematics at Boston University. Paul grew up in Sutton, Massachusetts, spent his undergraduate years at Brown University, and received his Ph.D. from Yale University. He has taught college mathematics for twenty-five years, mostly at Boston University. In 2001, he won the Northeast Section of the Mathematical Association of America's Award for Distinguished Teaching in Mathematics. He has coauthored or contributed chapters to four different textbooks. His main area of mathematical research is complex analytic dynamical systems and the related point sets, Julia sets and the Mandelbrot set. Most recently his efforts have focused on reforming the traditional differential equations course, and he is currently heading the Boston University Differential Equations Project and leading workshops in this innovative approach to teaching differential equations. When he becomes exhausted fixing the errors made by his two coauthors, he usually closes up his CD store and heads to the golf course with his caddy, Glen Hall.

Robert L. Devaney

Robert L. Devaney is Professor of Mathematics at Boston University. Robert was raised in Methuen, Massachusetts. He received his undergraduate degree from Holy Cross College and his Ph.D. from the University of California, Berkeley. He has taught at Boston University since 1980. His main area of research is complex dynamical systems, and he has lectured extensively throughout the world on this topic. In 1996 he received the National Excellence in Teaching Award from the Mathematical Association of America. When he gets sick of arguing with his coauthors over which topics to include in the differential equations course, he either turns up the volume of his opera CDs, or heads for waters off New England for a long distance sail.

Glen R. Hall

Glen R. Hall is Associate Professor of Mathematics at Boston University. Glen spent most of his youth in Denver, Colorado. His undergraduate degree comes from Carleton College and his Ph.D. comes from the University of Minnesota. His research interests are mainly in low-dimensional dynamics and celestial mechanics. He has published numerous articles on the dynamics of circle and annulus maps. For his research he has been awarded both NSF Postdoctoral and Sloan Foundation Fellowships. He has no plans to open a CD store since he is busy raising his two young sons. He is an untalented, but earnest, trumpet player and golfer. He once bicycled 148 miles in a single day.