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Currently I'm taking the Differential Equations course at college, however the problem is the book used. I'll try to make my point clear, but sorry if this question is silly or anything like that: the textbook used (William Boyce's book) seems to assume that the reader doesn't familiarity with abstract math, so it lacks that structure of presenting motivations, then definitions, then theorems and corolaries as we see in books like Spivak's Calculus or Apostol's Calculus.

I've already seem a question like this on Math Overflow, however unfortunatelly some people felt offended somehow and said: "are you saying that Boyce is easy? It doesn't matter if you know how to prove things, you must learn to compute", and my point is not that: Apostol and Spivak teaches how to compute also, however since their books are aimed to mathematicians they take care to build everything very fine and their main preocupation is indeed in the theoretical aspects.

I really don't like the approach of: well, in some strange way we found that this equation works, so memorize it, know how to compute things and everything is fine. I really want to understand what's going on, and until now I didn't find this possible with Boyce's book (certainly there are people that find this possible, but I'm used to books like Spivak's and Apostol's, so I don't really do well with books like Boyce's).

I've already seem Arnold's book on Differential Equations, but the prerequisites for reading it are bigger: he uses diffeomorphisms many times and although I'm currently also studying differential geometry, I don't feel yet comfortable on reading a book like this one.

Can someone recommend a book that covers ordinary differential equations, systems of differential equations, partial differential equations, and so on, but that can be read without much prerequisite, and that still has the structure of a book of mathematics? And when I say "structure of book of mathematics" is being like Spivak's and Apostol's books: not mixing up definitions, theorems and examples inside histories of how the theory developed. Since I'm a student of Mathematical Physics, of course motivations and examples from Physics are welcome, but not all mixed up in the text.

In truth I don't really believe that a book like the one I described exists (if a book on this topic is good in my point of view, I believe it'll have a lot of prerequisites). Anyway I hope that I don't get misunderstood in what's my doubt, and really sorry if this question is silly in someway.

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Try Birkhoff/Rota and Hirsch/Smale (I actually don't know the latest edition with Devaney).

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    $\begingroup$ Hi @TedShifrin, just for confirmation: Birkhoff/Rota's book is the "Ordinary Differential Equations" and Hirsch/Smale's book is the "Differential Equations, Dynamical Systems, And Linear Algebra" ? Thanks very much for your help! $\endgroup$ – user1620696 May 24 '13 at 1:25
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    $\begingroup$ Yup. Both not plug-and-chug, written by top-notch math scholars. $\endgroup$ – Ted Shifrin May 24 '13 at 1:26
  • $\begingroup$ Boyce and DiPrima is very different material wise to Devaney et al. The latter focuses a lot on planar systems and doesn't cover what should be covered in a first course I think. Plus it goes into other topics. However, I took a class in Advanced ODE's (which was a first course) and we actually used both. Boyce and DiPrima was used rather quickly (Chap.1-5 in like 2 weeks) and then we spent the rest of the semester with Devaney (Chap.1-7 & 17). $\endgroup$ – user70962 May 24 '13 at 1:39
  • $\begingroup$ @BryanUrízar Devaney's book is very different from Hirsch/Smale. It is systems-oriented, but not highly rigorous at all. We use it in our intro course here that's aimed at engineers and general science majors. $\endgroup$ – Ted Shifrin May 24 '13 at 1:42
  • $\begingroup$ @TedShifrin, I took a look on Birkhoff/Rota and it seems really good indeed, the kind of book I like to work with. However for my course I also need the basics of partial differential equations which this apparently don't cover. Can you also recommend one about this topic? $\endgroup$ – user1620696 May 24 '13 at 1:50
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"Theory of Ordinary Differential equations" by Coddington. If you find the previous one advanced, there is "An Introduction to Ordinary Differential Equations" by Coddington too. The first one is very complete and have a lot of things that I have never seen in other books. The second one is very basic (there is almost no qualitative theory), but proves the validity of some techniques usually used to solve ordinary differential equations. They also gives some explicit formulas for solution in special cases.Carmen Chicone's book "Ordinary Differential Equations with Applications" is good too if you want to study better the qualitative theory.

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I would recommend Differential Equations, Mechanics, and Computation You can read first pages to figure out whether it works for you. The book is 1) introductory, 2) rigorous, 3) with emphasis to physics.

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1). Try Verhulst's book on nonlinear differential equations. While the focus of this book is nonlinear ODEs, it does have a chapter on linear equations. It is a nice middle between the formula-computation based ODE books, and something like Arnold's ODE book.

2). For a more comprehensive but still accessible book, try the one by L. Perko. This is a gem.

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You could try the book by Tenenbaum. I think it may be partially in the direction you want, but not completely. Still some mixing of examples and the like, meant for motiviation. But rather thorough. And lots of proofs, albeit I won't say they try for Rudin style precision. IOW, definitely not a proof thereom correlary monograph. But pretty thorouogh and lots of proofs. Seems to be liked by a lot of people frustrated with diPrima. Very cheap buy too.

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