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I have no knowledge of differential equations, but I have the background in differential geometry/topology and analysis that one acquires in a PhD program. I.e. I have a foundational knowledge of Lie groups (roughly equivalent to Knapp's book), Riemannian geometry (roughly equivalent to do Carmo's book) and a similar foundational knowledge of real, complex and functional analysis. I'm looking for a book on differential equations to read in my spare time, but I'm finding it difficult to find one that is written with the level of care and detail that I look for in a textbook, the kind of care and detail one finds in John Lee's books, for instance. In many books I've looked at, symbols will be displayed in equations without precise definitions, functions will be mentioned without stating their domain and co-domain, and the exposition suffers from many other similar deficiencies in clarity.

Is there a carefully written DE book for someone with some mathematical maturity?

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    $\begingroup$ math.stackexchange.com/questions/757007/… $\endgroup$ – Turing Nov 20 '16 at 16:14
  • $\begingroup$ math.stackexchange.com/questions/243428/… $\endgroup$ – Turing Nov 20 '16 at 16:15
  • $\begingroup$ Seems like the obvious start is any undergraduate DE book. Based on your mathematical maturity you shouldn't take long to read through one. From there you should move to a more advanced text. And there are many. $\endgroup$ – Squirtle Nov 20 '16 at 16:17
  • $\begingroup$ @AlanTuring thanks, I should have mentioned that did see the related questions and I read through them. I thought my question was different enough to merit a new post, but maybe it could be seen as a duplicate. $\endgroup$ – Tim kinsella Nov 20 '16 at 16:19
  • $\begingroup$ @Timkinsella I could suggest you to talk with some professor too. Are you enrolled in some University? Well even if you are not, take a tour at some professor's office and have some talk! :D It's always beautiful. $\endgroup$ – Turing Nov 20 '16 at 16:21
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Here are two books to read:

  • Vladimir Arnold, ODE
  • Hirsch and Smale, Differential Equations, Dynamical Systems, and Linear Algebra, first edition

These are, arguably, two best introductory ODE books that are well suited for a good graduate course. The first one is very much intuitive, with many illustrations and sometimes lack of technical details. Which is more important, this book uses more modern language compare to many other textbooks, in particular, from the very beginning the discussion proceeds in terms of flows, transformation groups, and vector fields. The second book has a similar flavor, but written in a completely different style, with inclusion of many many mathematical details that are often left omitted. It also has a lot of very manageable exercises, which directly test the understanding.

Taken together, these two books will give you a very solid foundation in ODE, and also show connections to many other fields (differential geometry, Lie groups, analysis, etc).

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