Elementary Differential Equations and Boundary Value Problems delivers what it promises; a set of elementary differential equations and the techniques used to solve them. This book is replete with examples and has numerous problems to solve along with the book. Each chapter has an introduction to the problems at hand, an explanation of techniques used to solve the problems, the problems themselves, and references for further reading. Along the way, we are treated to little tidbits of trivia located in the footnotes. Most of the trivia is about famous mathematicians of the past and their contributions to the realm of mathematics or physics. This book expects a grounding in elementary calculus, but it still goes back and covers some of the topics that you should be familiar with. Since this edition of the book was printed in 1977, it doesn’t have that many pictures and very little color. Personally, I like it like this, since a lot of the images and graphs can get distracting. Since the book was originally printed in 1965 it might have some old terminology, but given the context I understood what was meant.
The book is divided into eleven main chapters, which are further subdivided into sections. These chapters are as follows;
Chapter 1 is merely an overview and introduction. It talks about what differential equations are, and the history that they have.
Chapter 2 is called First Order Differential Equations. Not much to say about this one. It starts with Linear Equations and goes on to Homogeneous Equations.
Chapter 3 is called Second Order Linear Equations.
Chapter 4 is called Series Solutions Of Second Order Linear Equations.
Chapter 5 follows Higher Order Linear Equations.
Chapter 6 discusses the Laplace Transform.
Chapter 7 discusses Systems of First Order Linear Equations.
Chapter 8 discusses Numerical Methods. This chapter probably needs an explanation. It starts with the Euler or Tangent Line Method, goes on to the error involved in it and improves on it. The following sections cover the Runge-Kutta Method and some other methods.
Chapter 9 is Nonlinear Differential Equations and Stability.
Chapter 10 is Partial Differential Equations and Fourier Series.
Chapter 11 is Boundary Value Theorems and Sturm-Liouville Theory.
Since this is a textbook, it contains a suggested syllabus for a classroom setting, assuming that you have a single semester of three hour classes.
All in all, this was a good book. It was written in such a way that it explained the terminology and didn’t go too far over my head. The main problem I have with advanced mathematics is that I only got up to Calculus II, and I don’t think I did too well in that case anyway. Being an autodidact is hard sometimes. Nonetheless, the book was quite good and written in a manner that I enjoyed. ( )

Another tried and true reference. My only regret is that I never owned a copy of it. (Managed to get through undergrad Diff-Eq by borrowing one from a friend.) ( )

Definitely did not like using this book. I found it hard to extract the vital information from it for tests and HW problems. It was not too clear with it's presentation though it covered a lot of information. ( )

This book is comprehensive and the concepts are well explained. I'm not sure if it's really a "mathemtatical" treatment of ODE's though. There's no proper proofs of existence and uniqueness, and also nothing on dependence on initial conditions and things like that.