Theory of Differential Equations Classical and Qualitative

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 pointPto pointQa 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 ofMathematicsfor 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 introdKelley, Walter G. is the author of 'Theory of Differential Equations Classical and Qualitative', published 2003 under ISBN 9780131020269 and ISBN 0131020269.