Confused about the theory behind Frobenius' Method

[nonequilibrium]

Hello,

My course is a bit vague on this topic.

1 First of all it states: if the differential equation A(x)y''(x) + B(x)y'(x) + C(x)y(x) = 0 with A, B, C analytical has A(a) = 0 but B(a) and C(a) NOT both zero, then a is a singular point and we can't solve it with a power series.
Now: what would it change if A(a) = B(a) = C(a) = 0? In other words:
Don't we need Frobenius for
$$x^2 y''(x) + x^2 y(x) = 0$$?
You might say "you don't, cause you can just divide the x away for x not zero" then well... I ask: can't we do exactly the same in all the cases where we use Frobenius?
(NB: I just tried to solve it with Frobenius but got the zero function as the solution...)

2 Then the course simply states the theorem that the differential equation
$$y''(x) + \frac{p(x)}{x}y'(x) + \frac{q(x)}{x^2}y(x) = 0$$
with p and q analytical around zero and R and r the (real) solutions of the indicial equation, then for x > 0 for R (> r) you certainly have the solution
$$y_1(x) = x^R \sum a_n(R) x^n$$
(it also says something about a second solution, but that's let no go into that here)

Now it briefly says that it only works for x > 0 because somewhere we choose to define the root of a complex number for theta in ]-pi,pi[ and we showed that that definition cannot be defined continuously in pi, but I don't really understand how any of that ties into the Frobenius method and where you make that choice in the solution of the Frobenius method.

3 I don't understand the importance of the Frobenius method: why go through so much trouble to get a power-series-esque solution around zero if we can get a regular power series around any number as close to zero as we'd like? (I'm not trashing the method, I'm just trying to understand its importance/significance!)

Thank you very much(!)

 

[HallsofIvy]

Specifically, what Frobenius' method says if that A(x)y"+ B(x)y'+ C(x)= 0 has a "regular singlular point" at x= a (and so can be solve by "Frobenius' method") if and only if
$$\displaytype\lim_{x\to a}\frac{A(x)}{C^2(x)}$$
and
$$\displaytype\lim_{x\to a}\frac{B(x)}{C(x)}$$
exist.

You might think of the "Euler-Cauchy" equation:
[tex]Ax^2y"+ Bxy'+ Cy= 0[/itex]
as the "boundary" between regular and irregular singular points.