Ordinary Differential Equations: Basics and Beyond (Texts in Applied Mathematics, 65) by David G. Schaeffer (PDF)

Description

This book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory, including bifurcation theory (but not chaos). While proofs are rigorous, the exposition is reader-friendly, aiming for the informality of face-to-face interactions. A unique feature of this book is the integration of rigorous theory with numerous applications of scientific interest. Besides providing motivation, this synthesis clarifies the theory and enhances scientific literacy. Other features include: (i) a wealth of exercises at various levels, along with commentary that explains why they matter; (ii) figures with consistent color conventions to identify nullclines, periodic orbits, stable and unstable manifolds; and (iii) a dedicated website with software templates, problem solutions, and other resources supporting the text (www.math.duke.edu/ode-book). Given its many applications, the book may be used comfortably in science and engineering courses as well as in mathematics courses. Its level is accessible to upper-level undergraduates but still appropriate for graduate students. The thoughtful presentation, which anticipates many confusions of beginning students, makes the book suitable for a teaching environment that emphasizes self-directed, active learning (including the so-called inverted classroom).

User’s Reviews

Editorial Reviews: From the Back Cover This book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory, including bifurcation theory (but not chaos). While proofs are rigorous, the exposition is reader-friendly, aiming for the informality of face-to-face interactions. A unique feature of this book is the integration of rigorous theory with numerous applications of scientific interest. Besides providing motivation, this synthesis clarifies the theory and enhances scientific literacy. Other features include: (i) a wealth of exercises at various levels, along with commentary that explains why they matter; (ii) figures with consistent color conventions to identify nullclines, periodic orbits, stable and unstable manifolds; and (iii) a dedicated website with software templates, problem solutions, and other resources supporting the text.Given its many applications, the book may be used comfortably in science and engineering courses as well as in mathematics courses. Its level is accessible to upper-level undergraduates but still appropriate for graduate students. The thoughtful presentation, which anticipates many confusions of beginning students, makes the book suitable for a teaching environment that emphasizes self-directed, active learning (including the so-called inverted classroom). About the Author David G. Schaeffer is Professor of Mathematics at Duke University. His research interests include partial differential equations and granular flow. John W. Cain is Professor of Mathematics at Harvard University. His background is in application-oriented mathematics with interest in applications to medicine, biology, and biochemistry.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐It’s a graduate level text. Excellent but hard. For example, you need to know advanced linear algebra, i.e. matrix analysis very well to understand Chapter 2: linear systems with constant coefficients.

⭐I regret that I had retired by the time this book came out, for I would have enjoyed teaching a first graduate ODE course from it. The authors acknowledge that there is too much material for one semester, and give a plausible looking set of cuts that can be made while leaving a coherent presentation of the most crucial material.There are a great many end-of-chapter exercises, categorized as “core” (to the chapter), or as related to particular sections of the chapter, and at the end of each set, some exercises labeled as “Ph.D”. Following the exercises are some “Pearls of Wisdom” – interesting asides to the main topics. Some exercises involve computer simulation, using some excellent (and free) software.There are many specific examples – far beyond the usuals of van der Pol, Duffing, Lorenz, and other famous names. Already in chapter one we have mass-spring systems, a pendulum, a cantilever beam, frictional forces, electrical circuits, and the Lotka-Volterra equations. van der Pol, for example, is revisited several times later in the book as new methods are introduced allowing more sophisticated results. There are many additional informal but useful remarks, a few even eliciting a smile. The authors have purposely designed the book for self-study as well as classroom use. A standard “advanced calculus”, or “undergraduate analysis” course is sufficient preparation, so that an undergraduate who has had such a course, and gotten an A (!), should be able to handle the material, though it will be a lot of work. I highly recommend this book for anyone wishing to learn about ordinary differential equations at a graduate level.

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