I teach at a community college, so I teach all the physics courses, and
some others as well. One of those "others" is the Ordinary Differential
Equations class. I teach that because none of our mathematics faculty
remember any of it, and admit that they can't.
This semester I had someone ask me a question I didn't know the answer
to nor can I find it anywhere; and, none of my colleagues in the math
department here can help me. So, I'm hoping for help from any of you...
The question I was asked came from the series solution portion of the
class. (As a quick reminder, these types of solutions give us Hermite
polynomials, Airy functions, Legendre functions, Bessel functions, etc.)
We can use the "ordinary point method" for most first order ordinary
differential equations, constant coefficient higher order equations, and
any non-constant coefficient problems that classify as being solved at an
ordinary point. For non-constant coefficient problems that are not
ordinary points, if the next level of test is performed and it classifies
as a "regular singular point" we can solve the problem using the Method of
Frobenius.
One very bright student asked -- could we just use the Method of Frobenius
for any of the solutions? I tried re-doing all of my examples I gave in
class using nothing but Method of Frobenius, and they all worked! However,
examples are not a proof. Do any of you know of a proof that shows the
Method of Frobenius could be used for anything other than just "regular
singular point" problems?