Second Course on ODEs – Self Study – Book Recommendation

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I'm looking for a book that would serve as a a second course on Ordinary Differential Equations. The book should be theoretically-oriented and for self study. I just finished the book by George Simmons – Differential Equations with Historical notes and Applications and in all honestly didn't really like it, finished it for the sake of completeness but it isn't what I was looking for.
I would like the book to contain a decent number of exercises after each section/chapter and, I stress, to be a theoretical approach to the topic. I originally started with Birkhoff and Rota's ODEs but one of the requierements was basic knowledge with complex functions, which I haven't studied yet.
I have knowledge of Calculus I (Spivak and Apostol), Calculus II (Apostol), Linear Algebra (Friedberg), Abstract Algebra (Herstein). Also, I will be finishing Introduction to Analysis (Mattuck) this week and will be starting with Rudin's Principles of Mathematical Analysis afterwards.
I have added the authors of the books, all of which I've liked, I used to study the previous topics for reference.
I'm looking for a book that I will be able to complete with the current knowledge that I have. I don't know if learning complex functions first, which I beleive I can take after finishing Rudin's Analysis, and then going back to Birkhoff's would be an option, but if it is I'm open to it too.

Thanks in advance

Best Answer

I will give you a recommendation of three texts on ordinary differential equations that I took from the "theorem-proof-proposed problems" approach.

However, I am sure that here at MathSE there are many experts who will be able to give you a better recommendation.

  1. Barbu, V. (2016). Differential Equations. (First edition). Cham, Switzerland: Springer.

  2. Hirsch, M. & Smale, S. & Devaney, R. (2013). Differential Equations, Dynamical Systems and an Introduction to Chaos. (3rd Edition). New York, USA: Academic Press.

  3. Arnold, V. (2006). Ordinary Differential Equations. (First edition). Berlin, Germany: Springer.

I highly recommend the first text, since it is a book that deals with the study of differential equations in a very formal way. However, it requires the knowledge of solid concepts in real and complex analysis in addition to a previous course in differential equations with a solid background. It seems to me that if you have studied from Simmons' book of differential equations, you will have no trouble adding solid analytical knowledge to that.

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