[Math] Good differential equations text for undergraduates who want to become pure mathematicians

ca.classical-analysis-and-odesdifferential equationsreference-requestsoft-questiontextbook-recommendation

Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate version of the usual graduate course at my university), analysis, model theory, and number theory. That is, I have taken enough 'abstract' (proof-based) mathematics courses to fall in love with the subject and decide to pursue it as a career.

However, I have been putting off taking a required ordinary differential equations course (colloquially referred to as 'calc 4', though this seems inappropriate) which will likely be very computational and designed to cater to the overpopulation of engineering students at my university.

So my question is, for someone who might have to actually concern themselves with the theory behind the 'rules' and theorems which will likely go unproven in this low-level course (likely of questionable mathematical content), what might be a decent supplementary text in ODE? That is, something substantive to counter-balance the 'ODE for students of science and engineering'-type text I will have to wade through. I want to study algebraic geometry further (I have gone through Karen Smith's text and the first part of Hartshorne), so something which goes from basic material through differential forms and related material would be nice.

Thanks! (and yes, it's embarrassing that I still haven't taken the 200-level ODE course, but I have been putting it off in favor of more interesting/rigorous courses… but now there's that whole graduation requirements issue).
–Lambdafunctor

Best Answer

Maybe I am reading too much into your pseudonym and your partly apologetic and partly condescending comments about the course you are going to take, but please,

Don't disparage the "rules" and computational aspects of differential equations.

Firstly, it is a beautiful subject with direct scientific origin and arguably most applications (save only calculus, perhaps) of all the courses you'd ever take. Secondly, these scientific connections continue to motivate and shape the development of the subject. Thirdly, rigor and abstraction are not substitutes for the actual mathematical content. Bourbaki never wrote a volume on differential equations, and the reason, I think, is that the subject is too content-rich to be amenable to axiomatic treatment. Finally, I've taught students who were gung-ho about rigorous real analysis, Rudin style, but couldn't compute the Taylor expansion of $\sqrt{1+x^3}.$ Knowing that the Riemann-Hilbert correspondence is an equivalence of triangulated categories may feel empowering, but as a matter of technique, it is mere stardust compared with the power of being able to compute the monodromy of a Fuchsian differential equation by hand.

Having forewarned you, here are my favorite introductory books on differential equations, all eminently suitable for self-study:

  • Piskunov, Differential and integral calculus
  • Filippov, Problems in differential equations
  • Arnold, Ordinary differential equations
  • Poincaré, On curves defined by differential equations
  • Arnold, Geometric theory of differential equations
  • Arnold, Mathematical methods of classical mechanics

You will find a lot of geometry, including an excellent exposition of calculus on manifolds, in the right context, in Arnold's Mathematical methods.

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