The Theory of Differential Equations: Classical & Qualitative

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9780131020269: The Theory of Differential Equations: Classical & Qualitative

This book introduces many of the important topics associated with modern and classical approaches to ordinary differential equations. Provides “refresher course” in differential equations. For mathematicians.

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Differential equations first appeared in the late seventeenth century in the work of Isaac Newton, Gottfried Wilhelm Leibniz, and the Bernoulli brothers, Jakob and Johann. They occurred as a natural consequence of the efforts of these great scientists to apply the new ideas of the calculus to certain problems in mechanics, such as the paths of motion of celestial bodies and the brachistochrone problem, which asks along which path from point P to point Q a frictionless object would descend in the least time. For over 300 years, differential equations have served as an essential tool for describing and analyzing problems in many scientific disciplines. Their importance has motivated generations of mathematicians and other scientists to develop methods of studying properties of their solutions, ranging from the early techniques of finding exact solutions in terms of elementary functions to modern methods of analytic and numerical approximation. Moreover, they have played a central role in the development of mathematics itself since questions about differential equations have spawned new areas of mathematics and advances in analysis, topology, algebra, and geometry have often offered new perspectives for differential equations.

This book provides an introduction to many of the important topics associated with ordinary differential equations. The material in the first six chapters is accessible to readers who are familiar with the basics of calculus, while some undergraduate analysis is needed for the more theoretical subjects covered in the final two chapters. The needed concepts from linear algebra are introduced with examples, as needed. Previous experience with differential equations is helpful but not required. Consequently, this book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students.

The first chapter contains some basic concepts and solution methods that will be used throughout the book. Since the discussion is limited to first-order equations, the ideas can be presented in a geometrically simple setting. For example, dynamics for a first-order equation can be described in a one-dimensional space. Many essential topics make an appearance here: existence, uniqueness, intervals of existence, variation of parameters, equilibria, stability, phase space, and bifurcations. Since proofs of existence-uniqueness theorems tend to be quite technical, they are reserved for the last chapter.

Systems of linear equations are the major topic of the second chapter. An unusual feature is the use of the Putzer algorithm to provide a cc9nstructive method for solving linear systems with constant coefficients. The study of stability for linear systems serves as a foundation for nonlinear systems in the next chapter. The important case of linear systems with periodic coefficients (Floquet theory) is included in this chapter.

Chapter 3, on autonomous systems, is really the heart of the subject and the foundation for studying differential equations from a dynamical viewpoint. The discussion of phase plane diagrams for two-dimensional systems contains many useful geometric ideas. Stability of equilibria is investigated by both Liapunov's direct method and the method of linearization. The most important methods for studying limit cycles, the Poincare-Bendixson theorem and the Hopf bifurcation theorem, are included here. The chapter also contains a brief look at complicated behavior in three dimensions and at the use of Mathematics for graphing solutions of differential equations. We give proofs of many of the results to illustrate why these methods work, but the more intricate verifications have been omitted in order to keep the chapter to a reasonable length and level of difficulty.

Perturbation methods, which are among the most powerful techniques for finding approximations of solutions of differential equations, are introduced in Chapter 4. The discussion includes singular perturbation problems, an important topic that is usually not covered in undergraduate texts.

The next two chapters return to linear equations and present a rich mix of classical subjects, such as self-adjointness, disconjugacy, Green's functions, Riccati equations, and the calculus of variations.

Since many applications involve the values of a solution at different input values, boundary value problems are studied in Chapter 7. The contraction mapping theorem and continuity methods are used to examine issues of existence, uniqueness, and approximation of solutions of nonlinear boundary value problems.

The final chapter contains a thorough discussion of the theoretical ideas that provide a foundation for the subject of differential equations. Here we state and prove the classical theorems that answer the following questions about solutions of initial value problems: Under what conditions does a solution exist, is it unique, what type of domain does a solution have, and what changes occur in a solution if we vary the initial condition or the value of a parameter? This chapter is at a higher level than the first six chapters of the book.

There are many examples and exercises throughout the book. A significant number of these involve differential equations that arise in applications to physics, biology, chemistry, engineering, and other areas. To avoid lengthy digressions, we have derived these equations from basic principles only in the simplest cases.

We would like to thank Deborah Brandon, Ross Chiquet, Valerie Cormani, Lynn Erbe, Kirsten Messer, James Mosely, Mark Pinsky, Mohammad Rammaha, and Jacob Weiss for helping with the proof reading of this book. We would like to thank Lloyd Jackson for his influence on Chapters 7 and 8 in this book. We would also like to thank Ned Hummel and John Davis ,,for their work on the figures that appear in this book. Allan Peterson would like to thank the National Science Foundation for the support of NSF Grant 0072505. We are very thankful for the great assistance that we got from the staff at Prentice Hall; in particular, we would like to thank our acquisitions editor, George Lobell; the production editor, Jeanne Audino; editorial assistant, Jennifer Brady; and copy editor, Patricia M. Daly, for the accomplished handling of this manuscript.

Walter Kelley
Allan Peterson

Review:

From the reviews of the second edition:

“A very good book on Differential Equations. It is the kind of book I would use in the classroom as well as recommend to a student for independent study. I can see it used as textbook for a course in Differential Equations ... or in a one semester course for math majors. ... instructors would enjoy teaching from this book, and that students would be able to study from it ... at a good pace. And they would learn a lot about differential equations.” (Florin Catrina, The Mathematical Association of America, November, 2010)

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