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I have two problem collections I am currently working through, the "Berkeley Problems in Mathematics" book, and the first of the three volumes of Putnam problems compiled by the MAA. These both contain many problems on basic differential equations.

Unfortunately, I never had a course in differential equations. Otherwise, my background is reasonably good, and I have knowledge of real analysis (at the level of baby Rudin), basic abstract algebra, topology, and complex analysis. I feel I could handle a more concise and mathematically mature approach to differential equations than the "cookbook" style that is normally given to first and second year students. I was wondering if someone to access to the above books that I am working through could suggest a concise reference that would cover what I need to know to solve the problems in them. In particular, it seems I need to know basic solution methods and basic existence and uniqueness theorem. On the other hand, I have no desire to specialize in differential equations, so a reference work like V.I Arnold's book on ordinary differential equations would not suit my needs, and I certainly don't have any need for knowledge of, say, the Laplace transform or PDEs.

To reiterate, I just need a concise, high level overview of the basic (by hand) solution techniques for differential equations, along with some discussion of the basic uniqueness and existence theorems. I realize this is rather vague, but looking through the two problem books I listed above should give a more precise idea of what I mean. Worked examples would be a plus. I am very unfamiliar with the subject matter, so thanks in advance for putting up with my very nebulous request.

EDIT: I found Coddington's "Intoduction to Ordinary Differential Equations" to be what I needed. Thanks guys.

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My instructor used these notes for the lectures of a basic theory of ODE course. Personally, I'm not a huge fan of the notes, but it does cover the uniqueness and existence (e.g. Gronwall's inequality) theorems pretty well and has a good mixture of computational and more theoretical exercises. The notes are based off of Ordinary Differential Equations By Birkhoff and Rota

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I like Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems. It has been around a long time and has been through many editions. You don't need the most recent one. You can probably find some edition somewhere at a good price.

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