Differential Equations (Springer Undergraduate Mathematics Series)

For disclosure, I have to mention that I am a former student of Viorel Barbu. I have learned differential equations from the original Romanian edition, which I continue to have on my desk for quick references.I am a practicing mathematician, and in my work differential equations appear frequently. The subject of ordinary differential equations (ODEs) is taught in US colleges, I have taught it myself several times. I like the subject but I'm not crazy about teaching this undergraduate class since most available textbooks spend enormous amount of time on how to solve these equations (very important to know) and on few applications that appear in many repetitive guises. From this point of view such a class resembles a glorious long list of how-to's that rarely goes beyond 19th century mathematics. The situation in graduate schools is even worse. There are few departments that offer a rigorous class on ODEs. It is a pity since many abstract results about ODEs are needed in other areas such as differential geometry or dynamical system. The close relationship between ODEs and flows (or dynamical systems) is a mystery to most graduate students I have taught.There are many books that cover the abstract theory of ODEs but they end up spending a lot of time to specialized subjects. This is a book directed first and foremost to the user of ODEs and secondly, to people interested in specializing in this subject. It covers a broad range of subjects with an efficiency and clarity I have not seen in many other sources. Let me give you a taste of what's inside.The first chapter covers most of the classical examples that can be solved "by-hand". It does so on 30 pages or so and it covers many more classes of equations typically taught in your regular undergraduate class. The second chapter is devoted to the abstract results: existence, uniqueness, continuous dependence of initial data and parameters. The third chapter is devoted to linear ODEs and systems of linear ODEs.There is a chapter devoted to stability and optimal control and a chapter on the method of characteristics for solving first order partial differential equations.Throughout there are applications that are hard to find anywhere at this level such as: ODEs associated to monotone operators including certain multivalued ones, Lyapunov functions and their applications to optimal control, dynamic programing in the guise of Hamilton-Jacobi equations. The proofs are designed so they extend with minimal effort to infinite dimensional situations. (Barbu is famous for his contributions to these infinite dimensional situations.) All this in under 200 pages.Another great virtue of the book is the collection of many juicy homework-type problems that illustrate the richness of this subject.The right audience for this book would be advanced undergraduates or beginning graduate students that have solid knowledge of linear algebra and the analysis of functions with several variables.Here's one (true) anecdote about this book. As I said earlier, I have on my desk a copy of the Romanian edition. A few of my (non-Romanian speaking) friends that would stop by my office to ask questions about ODEs noticed that I would open this book, and I would immediately find an answer there. So they started asking me to lend them the book, and for weeks at a time it would travel in different offices, used by people that would not understand Romanian, but would still be able to catch the mathematical ideas.I'm very happy there is an English edition of this little gem. Now my old ruffled book is back for good on my desk. I am sure the English edition it will help many generations of users of ODEs who will find that it is still fresh after all these years.

It is doing the job. (Ronseal!)

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