Looking for a 2nd book on Ordinary differential equations I have recently completed Tenenbaum & Pollard's ODEs book and am looking for something more in depth.
Something about my background- I have completed Rudin's PMA, Apostol's Analysis, Hubbard's book on multivariable analysis, Axler's book on Linear algebra and Marsden & Hoffman's Complex analysis.
I Have Shortlisted some of the books I have found, so it would be really helpful if someone could tell me which book should I choose which I can complete in a reasonably short time and would benefit the most from?

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*Arnold's books on ODEs

*Hirsch Smale

*Hirsch Smale & Devaney

*Ince

*Coddington & Levinson

*Jack Hale

*Birkhoff & Rota

 A: These books are not all equivalent and what you should study depends on how your interests lean. I don't have experience with all of them but here's what I can say about the ones I know.

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*Arnol'd's Ordinary Differential Equations is a relatively advanced book on ODEs that emphasizes (like all of Arnol'd's books) the geometric aspects of ODEs and connections to mechanics. It adopts a higher-dimensional perspective on ODEs immediately, and you will mostly treat ODEs as flows of vector fields on $\mathbb{R}^n$ and later on manifolds. It was written for a Soviet university audience in mind, which would have had exposure to analysis, classical mechanics, and maybe differential geometry prior to the book.


*Hirsch-Smale and Hirsch-Smale-Devaney are two editions of the same book, with the latter de-emphasizing much of the extensive treatment of linear systems found in the former. The highlight here is the focus in the second half of the book on nonlinear autonomous systems. This topic appears in Arnol'd too but has a more concrete and applied bent in Hirsch-Smale(-Devaney). The books can be thought of as a more advanced version of Strogatz's Nonlinear Dynamics and Chaos, which is a very popular book on the same subject. Hirsch-Smale(-Davaney) is the more rigorous of the two books and goes into more detail with some selected models, but Strogatz is very good too and accessible to a less proof-oriented audience.


*Coddington & Levinson is a classic text on ODEs introduced in the 50s. It should be read with a good understanding of a first course in analysis, complex analysis, and linear algebra. It covers some topics which were significant topics of interest back then, namely linear differential equations with holomorphic/meromorphic coefficients and the asymptotic analysis of their solutions near the poles. This is a topic that is rarely taught in undergraduate ODEs nowadays, partly because the audience of undergraduate ODEs has shifted to an engineering focus. It also contains an account of eigenvalue problems, linear boundary-value problems, and Sturm-Liouville theory, which are important topics any mathematician should learn. It does contain some serviceable chapters on nonlinear autonomous systems but doesn't have the detailed treatment you'd expect from a book like Hirsch-Smale(-Devaney) or Strogatz that is devoted to the subject. Ince is fairly comparable to Coddington & Levinson but older.
I am less informed about the other books.
Depending on how comfortable you are with the prerequisites and how much time you have on your hands, you could feasibly read any of these in the space of a semester, and they're all worthwhile books. The distinction really does come down to what special topics you want to learn.
