Diff EQs: Power Series vs Frobenius Solutions: Difference?

[Saladsamurai]

This is a pretty general conceptual question. I was just doing some reviewing for a test, and it occurred to me that if I were not told specifically to use Frobenius method on an equation, I might try to Power series solve it and vice versa. Can we talk about the difference a bit?

We apply both methods to the equation of the form: y'' + p(x)y' +q(x)y = 0

I know that using the Frobenius method has something to do with the fact that we can develop a solution about a regular singular point. But when would a power series solution fail where a Frobenius would not? It seems like we are taking a differential equation whose solution cannot be found due to singularities and forcing a solution from it by multiplying through by x² and solving that equation instead.

Any thoughts on this?

 

[Trifis]

It's a pity, there has not been any response to this question so far, since this issue is never explained properly by any introductionary textbook of ODEs, to the best of my knowledge. I suppose that the answer requires some background on complex DEs.

I would also like to point out, that the method is not always needed EVEN if we are looking for a solution in a neighboorhood of x₀, where x₀ is a regular singular point of p and q. For example, I cannot see, why the ordinary series method for the following Bessel-like equation: x²y''-xy'+(1-x)y=0 would fail! It delivers the same recursion formula as the solution with Frobenius method in wiki http://en.wikipedia.org/wiki/Frobenius_method does.