One of the best written books on the subject I've seen. As long as you're prepared to follow his anaylsis line by line (no skimming at the back there) you can give yourself some serious mathematical muscle.

He derives the first type of elliptical integral from the motion of the pendulum at page 21 (on my elderly foreign-printed software version I've had for years). That's the level it's pitched at.

This is one of the best books out there for introductory ODEs. This book should be read by everyone who's interested in the physical sciences. It has many applications to physics, and that's one of the main reasons I've decided to take the course. By the way, I took the course at UCLA with professor Grossman (he reviewed the book below). He's a brilliant and a challenging mathematician, which comes at no surprise since he's a graduate of Caltech. Anyhow, I really recommend this book for all mathemticians and physicists. It's a great investment, since it can be a great reference in the future!

Each mathematic's book has its particularities. I had said before the Tanembaum book was the one better than I had on differential equations, considering its easiness reading , its methodology and its organization, but evidently these three elements can be present in many other books. This is the case of the present book, which I consider excellent. This is a great book, a very well achieved work . It introduces each element interesting the reader for the following topics, with exercises very well planned according to the exposed theory and with answers at the end of the book. Each chapter contains a historical note that will be extremely instructive and stimulants for the reader that doesn't feel pleasure for the exact sciences . Of special mention they are the chapter 6 (Some special functions of the mathematical physics), the chapter 8 (non lineal equations), and the chapter 9 (calculus of variations). The calculation of variations is an introduction to this branch, but I assure you that when reading it, you kept desires of treating other texts specialized in the topic like that of Sagan. You won't find in this book any complication, and you will be satisfied considering that it is a book for undergraduates, an excellent introduction to the differential equations.

Among the book of Tanembaum, Derry Grossman, D'prima, and the introductions of differential equations of C. R. Wylie and Kreiszig in their books of advanced mathematics for engineering, I keep this. If you plan to learn differential equations, if you want to have a really good text as introduction to the topic, then this is the book to buy, you won't lose your money.