| “Unavoidable” is a bit too strong I think. For an ODE course, what does the usual list of elementary methods really include? - Separation of variables. If one is fine with differentials (or their modern cousins differential forms), there isn’t much to explain here. - Linear equations solved with quasipolynomials. The only ODE-specific observation is that d/dx in the ( x^k e^x / k! ) basis is a Jordan block; the rest is the theory of the Jordan normal form, which makes interesting mathematical points (an embryonic form of representation theory) but exists entirely within linear algebra (even if it was motivated by linear ODEs historically). - Ricatti equations. Were always a mystery to me, but it appears they could also be called “projective ODEs” to go with linear ones and have pretty nice geometry behind them (even if, as you said, they were first discovered by brute force search). - Variation of parameters. Despite the mysterious appearance, this is simply the ODE case of Green’s method beloved in its PDE version by physicists and engineers. (This isn’t often included in textbooks, in fear of scaring students with Dirac’s delta, but Arnold does explain it, and IIRC Courant–Hilbert mentions it in passing as well.) - Integrating factors. Okay, I can’t really explain what that one means, even though it feels like I should be able to. Not that teaching it like this would make for a good course (too general, and ODEs ≠ methods for solving ODEs), but that’s essentially it, right? There are certainly other methods you could mention, and not unimportant ones (perturbation theory!.. -ries?), but this basically covers the standard litany as far as I can see. And it’s no haphazard collection of tricks—none of these is just pulling solutions out of a hat. (In the interest of changing things up and not spending an hour on a single comment, I will omit the barrage of references I’d usually want to include with this list, but I can dig them up if somebody actually wants them.) |
Here the first ODE course is half a semester. If you spend a week or two proving existence and unicity, you get one week to study each method and make a few examples and then you must change to next week trick.
Fourier/Laplace and other advanced stuff are in a more advanced course.
I never used perturbation theory for ODE. I've seen it for solving eigenvalues/eigenvector of operators in QN. But perhaps it's one tool I don't know.