Synopses & Reviews
Incorporating a modeling approach throughout, this exciting text emphasizes concepts and shows that the study of differential equations is a beautiful application of the ideas and techniques of calculus to everyday life. By taking advantage of readily available technology, the authors eliminate most of the specialized techniques for deriving formulas for solutions found in traditional texts and replace them with topics that focus on the formulation of differential equations and the interpretations of their solutions. Students will generally attack a given equation from three different points of view to obtain an understanding of the solutions: qualitative, numeric, and analytic. Since many of the most important differential equations are nonlinear, students learn that numerical and qualitative techniques are more effective than analytic techniques in this setting. Overall, students discover how to identify and work effectively with the mathematics in everyday life, and they learn how to express the fundamental principles that govern many phenomena in the language of differential equations.
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 willuse in their careers, and apply it effectively outside the classroom.
About 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.
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. Integration Factors for Linear Equations. Review Exercises for Chapter 1. Labs for Chapter 1. 2. FIRST-ORDER SYSTEMS. Modeling via Systems. The Geometry of Systems. Analytic Methods for Special Systems. Euler's Method for Systems. The Lorenz Equations. Review Exercises for Chapter 2. Labs for Chapter 2. 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. Review Exercises for Chapter 3. Labs for Chapter 3. 4. FORCING AND RESONANCE. Forced Harmonic Oscillators. Sinusoidal Forcing. Undamped Forcing and Resonance. Amplitude and Phase of the Steady State. The Tacoma Narrows Bridge. Review Exercises for Chapter 4. Labs for Chapter 4. 5. NONLINEAR SYSTEMS. Equilibrium Point Analysis. Qualitative Analysis. Hamiltonian Systems. Dissipative Systems. Nonlinear Systems in Three Dimensions. Periodic Forcing of Nonlinear Systems and Chaos. Review Exercises for Chapter 5. Labs for Chapter 5. 6. LAPLACE TRANSFORMS. Laplace Transforms. Discontinuous Functions. Second-Order Equations. Delta Functions and Impulse Forcing. Convolutions. The Qualitative Theory of Laplace Transforms. Review Exercises for Chapter 6. Labs for Chapter 6. 7. NUMERICAL METHODS. Numerical Error in Euler's Method. Improving Euler's Method. The Runge-Kutta Method. The Effects of Finite Arithmetic. Review Exercises for Chapter 7. Labs for Chapter 7. 8. DISCRETE DYNAMICAL SYSTEMS. The Discrete Logistic Equation. Fixed Points and Periodic Points. Bifurcations. Chaos. Chaos in the Lorenz System. Review Exercises for Chapter 8. Labs for Chapter 8. Hints and Answers. APPENDICES. Changing Variables. The Ultimate Guess. Complex Numbers and Euler's Formula.