Second Course on ODEs - Self Study - Book Recommendation

Question

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

Answer 1

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.

Answer 2

I would like to recommend L. Perko's book: Differential Equations and Dynamical Systems, 3rd ed to you. This is because it contains more modern and hot topics in dynamical system. P. Smith's book Nonlinear Ordinary Differential Equations is also a good book for studying nonlinear ODEs.