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Differential Equations / Edition 1

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Overview

Incorporating an innovative modeling approach, this book for a one-semester differential equations course emphasizes conceptual understanding to help users relate information taught in the classroom to real-world experiences. Certain models reappear throughout the book 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. Users will discover how to identify and harness the mathematics they will use in their careers, and apply it effectively outside the classroom.

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Editorial Reviews

From The Critics
A departure from the typical "cookbook" differential equations text, this book/CD-ROM package focuses on the formulation of differential equations and the interpretation of their solutions. To enhance understanding of solutions, equations are tackled from qualitative, numerical, geometric, and graphical perspectives. Other nontraditional features of the text include the incorporation of modeling throughout, and adoption of a dynamical systems perspective. The CD-ROM is new to this edition and contains demonstrations keyed to text exercises. It also includes computer-based tools for investigating the behavior of solutions of differential equations both numerically and graphically. Blanchard has taught college mathematics for 25 years. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780534352530
  • Publisher: Brooks/Cole
  • Publication date: 10/28/1998
  • Series: Mathematics Series
  • Edition description: Older Edition
  • Edition number: 1
  • Pages: 242

Meet the Author

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 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 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.

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Table of Contents

1 First-Order Differential Equations 1
1.1 Modeling via Differential Equations 2
1.2 Analytic Technique: Separation of Variables 20
1.3 Qualitative Technique: Slope Fields 36
1.4 Numerical Technique: Euler's Method 53
1.5 Existence and Uniqueness of Solutions 65
1.6 Equilibria and the Phase Line 76
1.7 Bifurcations 96
1.8 Linear Differential Equations 113
1.9 Changing Variables 123
Labs for Chapter 1 138
2 First-Order Systems 147
2.1 Modeling via Systems 148
2.2 The Geometry of Systems 165
2.3 Analytic Methods for Special Systems 183
2.4 Euler's Method for Systems 194
2.5 The Lorenz Equations 209
Labs for Chapter 2 216
3 Linear Systems 225
3.1 Properties of Linear Systems and the Linearity Principle 226
3.2 Straight-Line Solutions 250
3.3 Phase Planes for Linear Systems with Real Eigenvalues 266
3.4 Complex Eigenvalues 282
3.5 Special Cases: Repeated and Zero Eigenvalues 301
3.6 Second-Order Linear Equations 316
3.7 The Trace-Determinant Plane 333
3.8 Linear Systems in Three Dimensions 346
Labs for Chapter 3 362
4 Forcing and Resonance 369
4.1 Forced Harmonic Oscillators 370
4.2 Sinusoidal Forcing 385
4.3 Undamped Forcing and Resonance 397
4.4 Amplitude and Phase of the Steady State 409
4.5 The Tacoma Narrows Bridge 421
Labs for Chapter 4 431
5 Nonlinear Systems 437
5.1 Equilibrium Point Analysis 438
5.2 Qualitative Analysis 457
5.3 Hamiltonian Systems 470
5.4 Dissipative Systems 488
5.5 Nonlinear Systems in Three Dimensions 510
5.6 Periodic Forcing of Nonlinear Systems and Chaos 518
Labs for Chapter 5 535
6 Laplace Transforms 541
6.1 Laplace Transforms 542
6.2 Discontinuous Functions 554
6.3 Second-Order Equations 563
6.4 Delta Functions and Impulse Forcing 577
6.5 Convolutions 585
6.6 The Qualitative Theory of Laplace Transforms 594
Labs for Chapter 6 603
7 Numerical Methods 607
7.1 Numerical Error in Euler's Method 608
7.2 Improving Euler's Method 621
7.3 The Runge-Kutta Method 629
7.4 The Effects of Finite Arithmetic 640
Labs for Chapter 7 644
8 Discrete Dynamical Systems 647
8.1 The Discrete Logistic Equation 648
8.2 Fixed Points and Periodic Points 661
8.3 Bifurcations 670
8.4 Chaos 679
8.5 Chaos in the Lorenz System 687
Labs for Chapter 8 693
Appendices 699
A First-Order Linear Equations Revisited 700
B Complex Numbers and Euler's Formula 711
Hints and Answers 717
Index 777
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Sort by: Showing all of 3 Customer Reviews
  • Anonymous

    Posted May 22, 2002

    Try Something else

    I used this course in a differential equations course and found it disorganized and confusing. An entire section was built on the notion of 'guessing' solutions -- this was completely unnecessary as subsequent sections showed how to avoid the guessing process in solving ordinary d.e. Overall, there must be better introductions to this interesting subject.

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  • Anonymous

    Posted November 30, 2001

    Effective introduction

    The prose was uniformly easy to follow and not at all stilted. The presentation was generally clear but I agree with others that the pace was sometimes too slow and sometimes slightly too fast. The appendix included enough background on prerequisite topics like the complex exponential to make the book accessible to anyone with a good foundation in calculus. I personally did well in my course and feel I owe that success, in large part, to this book.

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  • Anonymous

    Posted August 21, 2001

    Best Introductory ODE Book Available!

    Nothing but good things to say about the book... written in a very reader friendly style without losing any necessary mathematical rigor.

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