I am a freshman (college) math student with semi-decent experience in rigorous maths: I've taught myself / taken courses in real analysis, abstract algebra, measure theory, linear algebra, and of course calculus up to and including the beautiful Stokes', Green's and Gauss' Theorems.
Due to some scheduling... things, I am set to take a proof-based course on PDEs (still with lots of computations) before I have taken a course on ODEs. I have spoken with the professor, who thinks that I should be ready to take it--and he has told me that ODEs is technically not a pre-requisite. That being said, I would like to take the time before that to teach myself ODEs to some capacity, at least to learn the basic theorems on existence and uniqueness, and then teach myself ODEs further concurrently with the PDE class.
While looking for a textbook, I stumbled across this rather popular old post on MO; the questioner asks for a ODE book that leans towards the theoretical. Many of the answers, and other sources, seem to concur that one who over-prioritizes rigor when learning ODEs a first time risks losing either intuition, or not learning the full usefulness of ODEs. (1)
That being said, I have purchased the books:
- Vladmir Arnol'd, Ordinary Differential Equations
- Jack K. Hale, Ordinary Differential Equations
Both are more 'theoretical' than some books I considered, like Tenenbaum and Pollard (which also seems good, but the thickness did put me off a little). Arnol'd seems rather legendary at instilling geometric intuition along with rigor, while Hale seems like an incredibly well-written theorem-proof style book geared more towards graduate students. Reading through both, though, I found that Hale's style easier to understand. A big bonus for me is that Hale talks about and proves uniqueness / existence in the first few pages, while Arnol'd leaves it to the Appendix.
But, I have been getting anxious that choosing to learn from Hale. Am I falling into the trap of statement (1)? Am I losing something by choosing the more 'rigorous' book? I comfort myself slightly by the fact that Hale's book was designed for an applied mathematics course (at Brown University), but still. These worries are compounded by the possibility that I may go into mathematical physics (with very heavy emphasis on the mathematical part). I am planning to teach myself some physics using Arnol'ds Mathematical Methods of Classical Mechanics-- so perhaps I will make up for the 'over-rigorousness' of my approach there?
I would be grateful to hear opinions on statement (1), and opinions specific to my situation from anyone who is familiar with Jack Hale's book.
Boyce and Diprima have very popular texts I have seen used in multiple universities. They have both passed away and Meade is now revising them. I believe there are a variety of editions now with different emphases.
Math is not merely theory but also techniques. Particularly because many ODE's cannot be solved analytically, numeric methods and computer applications are frequently emphasized. This gets into my passion- the intersection of computers and math. Maybe the question- "How can different computer systems arrive at significantly different conclusions to the same exercise?" has some interest. Also, which system to trust? What criteria do we even look at?
Personally, I usually buy older editions for financial reasons and because the texts usually haven't changed incredibly.
I'm surprised by your question though. Most universities have a full course of ODE for Math Majors and a separate ODE course for engineers which is less intensive and includes a week or so of linear algebra. Even in the introductory math major only courses beyond existence and uniqueness, I haven't seen a lot of theory only, mostly just technique. It sounds like you'll want to learn both techniques and add theory. The parts you are particularly interested aren't usually dealt with until after the first semester.