A book on the ins and outs of ordinary differential equations Please I want to request for recommendation of a book that can be used to have a good grasp of the concepts of ODE. Mainly the uniqueness, existence, stability and various proofs and applications of solution. I have this book by Ferdinand Verhulst, but it isn't detailed at all. And also can anyone help with Ordinary Differential Equations: Introduction and qualitative theory by Jane Cronin?
 A: Here are some good books:
Robinson - An Introduction to ODEs
Pollard - Ordinary Differential Equations
Boyce & DiPrima - Elementary Differential Equations
Arnold - Ordinary Differential Equations
Arnold is fairly advanced.
A: *

*The Qualitative Theory of Ordinary Differential Equations: An Introduction (Dover Books on Mathematics), Fred Brauer, John A. Nohel

*Qualitative Theory of Differential Equations (Dover Books on Mathematics), V. V. Nemytskii , V. V. Stepanov

*Ordinary Differential Equations: Qualitative Theory (Graduate Studies in Mathematics), Luis Barreira, Claudia Valls 

*Ordinary Differential Equations (Dover Books on Mathematics), Edward L. Ince

*An Introduction to Ordinary Differential Equations (Dover Books on Mathematics), Earl A. Coddington

*Schaum's Outline of Differential Equations, 3ed (Schaum's Outline Series), Richard Bronson, Gabriel Costa

*Qualitative Theory of Differential Equations, Zhang Zhi-fen, Ding Tong-Ren, Huang Wen-zao, Dong Zhen-xi

*Approaches to the Qualitative Theory of Ordinary Differential Equations Dynamical Systems and Nonlinear Oscillations Ding Tongren (Peking University, China)


You might also want to peruse several other postings on MSE regarding book recommendations in this particular area. Also, look in to Open Courseware (like MIT) and online notes as there is a lot of free material (some excellent materials) out there.
A: Oldie but goodie:  Ordinary Differential Equations by Jack K. Hale.  Originally published by John Wiley & Sons, Inc., New York, 1969.  Second edition published by Robert E. Krieger Publishing Company, New York, 1980.  Second edition re-publihed by Dover in 2009; available from them for a song.  ISBN-13:  978-0-486-47211-9; ISBN-10:  0-486-47211-6.  LCN QA372.H184 2009.  (The reason I give all this information is that I have the Dover edition in front of me right now; in fact, I carry it around.) Has thorough treatments of existence, uniqueness, stability, periodicity, integral manifolds.  Pretty advanced, I'd say.  One of my favorites.
Another short but sweet treatment is Witold Hurewicz's book on ODEs, can't remember the exact title right now, though I'm pretty sure it contains the phrase "Differential Equations".  About 100 pages, very clear and well done.
Arnold is great as well, as 5space mentioned in his answer.
If you want to see something really engaging, check out Poincare's New Methods.
Practically invents the modern theory . . .
Then there's always that really old favorite Principia by some Brit named Newton . . . ;)
A: Differential equations and dynamical systems by Lawrence perko might fit the bill for you. When I was learning theory of ODEs, I used this and Verhulst, along with some Arnold.
