From what I've seen when I tried to find an alternative book, this might be the best out there. For someone who isn't naturally mathematically intuitive, this book sure doesn't take it easy on me. I understand that Differential Equations is a higher level math course. But this book left a lot of questions unanswered. To get through my semester, I ended up purchasing the student solutions manual and two other differential equations books. I also made use of the free lectures on the MIT website. I couldn't have gone through this course (and this book) without the supplements!

It's sad to see that I couldn't find a more highly rated differential equations text on Amazon. If someone knows of one, let me know! I finished the course, but I really would like to have a better grasp. It really is an amazing and practical field of mathematics, and apparently lays the groundwork for a lot of application. It would be a shame if I never got to fully understand it because I couldn't get my brain wrapped around this book. Who knows? Maybe I'm just slow. ( )

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.