[gus_massa]

I was thinking in something similar. (I have no idea what a Ricatti equation is https://en.wikipedia.org/wiki/Riccati_equation ) Perhspas I should have said "half a dozen".

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.