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This question has been posted before, but I need book with specific qualifications. I do not need books for engineers, book that is centered around calculations and stuff. I need to find a book that is theoretical, proves the statements and has good presentation of the theoretical structure. I have had the book by Tenenbaum, I did not like it. I would be very very thankful if someone shared their knowledge with us about this matter.

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  • $\begingroup$ Could you highlight what themes you want to be covered in such book? $\endgroup$ – Evgeny Nov 19 '14 at 19:10
  • $\begingroup$ @Evgeny Sure. I want it to cover the basic definitions and geometric interpretation, osgood and peano theorems, arzela's theorem, picard method, euler lines, reduction of a system to first order equations, reduction to canonical form, and things like that. These are the things that came to my mind first. Something that explores theoretical foundations from the basics $\endgroup$ – Marion Crane Nov 19 '14 at 22:20
  • $\begingroup$ @primenumber57 I do not know a lot of book that would prove Osgood's theorem. One of them is Hartman, ODE which is basically a bible for researches in ODE, and covers pretty much what was known by 1960. But this is not a textbook, and it requires quite a good background to start reading. $\endgroup$ – Artem Nov 19 '14 at 23:35
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    $\begingroup$ And here is one more example, which comes to mind: a book for famous Russian mathematician: Ordinary Differential Equations, which does not cover that much, but what is covered, is covered with absolute rigor and detail. Even better if you read Russian and can pick up a last edition. $\endgroup$ – Artem Nov 19 '14 at 23:41
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    $\begingroup$ Apart from Petrovsky's, try also Arnold's and Pontryagin's books of the same titles, which are more geometric. If you really like concision, try Hurewicz. $\endgroup$ – Marius Kempe Nov 20 '14 at 17:39
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The best such book is Differential Equations, Dynamical Systems, and Linear Algebra.

You should get the first edition. In the second and third editions one author was added and the book was ruined.

This book suppose very little, but 100% rigorous, covering all the excruciating details, which are missed in most other books (pick Arnold's ODE to see what I mean). It covers a lot, with the emphasizes on the theory and not on the computational side. However, a lot of exercises will teach you also to compute.

I cannot find enough praise for this book.

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    $\begingroup$ Does the 2nd/3rd editions differ so much from the 1st? $\endgroup$ – Evgeny Nov 19 '14 at 19:20
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    $\begingroup$ @Evgeny Yes, there is very big difference. Most important, the level of rigor was reduced to make it possible to use this book in undergraduate "Nonlinear dynamics and chaos" courses. $\endgroup$ – Artem Nov 19 '14 at 19:51
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Here are a few more suggestions:

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You could always try Ordinary Differential Equations by Tenenbaum. It's very comprehensive and thorough.

http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=sr_1_1?ie=UTF8&qid=1416423343&sr=8-1&keywords=differential+equation+tenenbaum

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  • $\begingroup$ I have that book. I have read that. I did not like it tho $\endgroup$ – Marion Crane Nov 19 '14 at 18:57
  • $\begingroup$ Differential Equations: A First Course, Third Edition by Guterman. $\endgroup$ – Enigma Nov 19 '14 at 19:02

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