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I'm having trouble solving ordinary differential equations (DE's), because I don't understand the underlying theory, nor how to apply it.

For example, suppose I am solving such a DE. My techniques allow me to find the general solution on an interval $(a,b),$ and also on an interval $(b,c).$ Under what circumstances can I, by gluing pairs of solutions, obtain the general solution on the larger interval $(a,c)$ ? In particular, under what circumstances can I conclude that all solutions on $(a,c)$ can be obtained in this way?

Is there a book (article, lecture series etc.) that explains these kinds of things?

Here's what I'm looking for in a book.

  • Precisely stated definitions and theorems.
  • Examples of how to apply them in order to solve problems rigorously. (By "rigorous solution to a problem," let us mean a solution that not only gets the right answer, but also constitutes a proof that the answer is correct.)
  • An author with a keen sense for the difference between an argument that can easily be made rigorous, versus an argument that sounds convincing but which upon closer inspection is full of holes.

Here's what I don't (currently) need.

  • Formal proofs of the theorems.
  • PDE's (yet!).
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I think the older generation of DEs books may serve your interests better. Personally, I gained much from reading the text by Ritger and Rose and also Rabenstein. Theorems are at least sometimes stated in terms of explicit existence of $\delta$-neighborhoods etc... Boyce and Diprima is also a serious text. I feel your pain here, too many of the basic texts have been driven by the market to abdicate the actual statement of mathematics. Applications, qualitative analysis, really just an excuse to dumb down the books. See Boyce and Diprima... – James S. Cook Sep 8 '13 at 11:32
you'll see it has both attention to detail and qualititative analysis – James S. Cook Sep 8 '13 at 11:32
An introductory text that is very careful with correct and precise statements and justifications, but which isn't an upper-level "ODE theory text", is Differential Equations by Ralph Palmer Agnew. This book is also very well written and it is even quite humorous in places. – Dave L. Renfro Sep 11 '13 at 20:26
@DaveL.Renfro, thanks for the reference. My university doesn't have it so it may take me a while to check it out. I've stopped buying DE books outright after a few disappointments... ;) – goblin Sep 12 '13 at 0:50
If you have access to interlibrary loan, try to get a copy that way. Then you can decide whether you want to buy a used copy for yourself. This is one of those books that's too easy for advanced undergraduate theory courses, but probably has too many rigousous technical issues discussed to be widely used, so I don't believe it was ever a best selling text "back in the day", and now it's old enough that few people know about it anymore. – Dave L. Renfro Sep 12 '13 at 13:56
  • For ODE's, I recommend Ordinary Differential Equations by M. Tenenbaum, H. Pollard.
  • For PDE's I suggest you to have look at Elements of Partial Differential Equations by Ian N. Sneddon.
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Very good suggestions +1 – Amzoti Sep 8 '13 at 12:15
@Amzoti: Good Mornin :-) – S. Snape Sep 8 '13 at 12:17
Good morning and have a great day! Regards – Amzoti Sep 8 '13 at 12:18
Nice suggestions, indeed +1 – amWhy Sep 9 '13 at 0:10

and i recommend a book called

Williams E. Boyce and Richard C. DIPrima, Elementary differential equations and boundary value problems, John Wiley and sons, New York

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