I personally like Jack Hale's book titled "Ordinary Differential Equations". I am a 3rd year graduate student studying Hamiltonian systems and have needed to review/learn topics in ODE's from time to time, and Hale's book has stood out as the most valuable resource for me.

I found Hale's book to be most readable, well-organized, and informative book covering the basics of ODE's. The treatment of the most basic issue of ODE's, i.e. existence/uniquness, is extremely well-written.

A lot of people seem to like Arnold's ODE book, and although it is a good book, I've had much better luck learning from Hale's book. Except for introducing differential equations on manifolds, all the main topics in Arnold's book are a subset of those in Hale's book. Hale also covers topics such as the Poincare-Bendixson Theorem and gets into stable/unstable manifolds, neither of which are present in Arnold's book. The presentation on time-periodic systems and related stability issues is also much clearer in Hale's book.

Another book I occasionally look at is Smale and Hirsh's book. This book is much more elementary than Hales and Arnold's, but has a few nice examples, especially in the few chapters regarding applications. However, the early section on The Poincare map is horrible, and should be the last place a person goes to learn what the Poincare map is.

To recap, if I need to look something up I looked at:

Hale
Arnold
Smale and Hirsch

Every time Hale's book wins in terms of readability, depth, and being easy to navigate through.

Last, but not least: Hale's book is published by Dover, and is quite cheap.

I personally like Jack Hale's book titled "Ordinary Differential Equations". I am a 3rd year graduate student studying Hamiltonian systems and have needed to review/learn topics in ODE's from time to time, and Hale's book has stood out as the most valuable resource for me.

I found Hale's book to be most readable, well-organized, and informative book covering the basics of ODE's.

A lot of people seem to like Arnold's ODE book, and although it is a good book, I've had much better luck learning from Hale's book. Except for introducing differential equations on manifolds, all the main topics in Arnold's book are a subset of those in Hale's book. Hale also covers topics such as the Poincare-Bendixson Theorem and gets into stable/unstable manifolds, neither of which are present in Arnold's book. The presentation on time-periodic systems and related stability issues is also much clearer in Hale's book.

Another book I've I occasionally look at from time to time is Smale and Hirsh's book. This book is much more elementary than Hales and Arnold's, but has a few nice examples, especially in the few chapters regarding applications. However, the early section on The Poincare map is horrible, and should be the last place a person goes to learn what the Poincare map is.

Last, but not least: Hale's book is published by Dover, and is quite cheap.

I personally like Jack Hale's book titled "Ordinary Differential Equations". I am a 3rd year graduate student studying Hamiltonian systems and have needed to review/learn topics in ODE's from time to time, and Hale's book has stood out as the most valuable resource for me.

I found Hale's book to be most readable, well-organized, and informative book covering the basics of ODE's.

A lot of people seem to like Arnold's ODE book, and although it is a good book, I've had much better luck learning from Hale's book. Except for introducing differential equations on manifolds, all the main topics in Arnold's book are a subset of those in Hale's book. Hale also covers topics such as the Poincare-Bendixson Theorem and gets into stable/unstable manifolds, neither of which are present in Arnold's book. The presentation on time-periodic systems and related stability issues is also much clearer in Hale's book.

Another book I've look at from time to time is Smale and Hirsh's book. This book is much more elementary than Hales and Arnold's, but has a few nice examples, especially in the few chapters regarding applications. However, the early section on The Poincare map is horrible, and should be the last place a person goes to learn what the Poincare map is.

Last, but not least: Hale's book is published by Dover, and is quite cheap.