Is Frobenius' method generally useful?

Question

Regarding the differential equation $$ y'' + p(z)y' + q(z)y = 0,\quad z\in\mathbb{C}, $$ we can find solutions of the form $$ \sum_{n=0}^\infty c_n (z-z_0)^n, \quad c_n\in\mathbb{C}, $$ given that $p(z)$ and $q(z)$ are analytic in $z=z_0$. Here $|z-z_0|<R_1$ for some $R_1>0$ must hold. Suppose $p(z)$ and $q(z)$ aren't analytic at $z=z_0$, but $(z-z_0)p(z)$ and $(z-z_0)^2 q(z)$ are, then we can find solutions of the form $$ \sum_{n=0}^\infty c_n (z-z_0)^{n+r}, \quad c_n\in\mathbb{C}, $$ where $r$ satisfies $r(r-1)+[zp(z)]_{z=z_0}r+[z^2q(z)]_{z=z_0}=0$. Here $|z-z_0|<R_2$ for some $R_2>0$ must hold. The latter is Frobenius' method.

Now to illustrate my question suppose $z_0=0$ is a singular point of $p(z)$ but not of $zp(z)$ so we would apply Frobenius' method. Then suppose it turns out that $R_2$ is really small. In my understanding we could also have looked for solutions around some analytic point. Suppose $z_0=2$ is analytic, then we would've looked for solutions of the form $\sum_{n=0}^\infty c_n (z-2)^n$ and then maybe the corresponding $R_1$ would have turned out to be more satisfactory.

So if what I said in the above is right, my question is: can we know beforehand if applying Frobenius' method is useful?

Answer 1

Frobenius method works well even in the case when $x_0$ is an ordinary point of $p(x)$ and $q(x)$, where these are coefficients of $y"(x)+p(x)y'(x)+q(x)y=0$.

Specifically,$p(x)=p_0+p_1.x+p_2.x^2+...$ and $q(x)=q_0+q_1.x+q_2.x^2+.....$ have power series expansions as they are analytic.

In this case, the Frobenius series of the form with ($x_0=0$ say):

$y(x)=x^m(a_0+a_1.x+a_2.x^2+....)$ reduces to a power series if $m=0$, which is indeed a root of your indicial equation $m(m-1)+p_0.m+q_0=0$.

Edit:- Though you can find a power series solution in the neighborhood of $x_0=2$, but the solution $y(x)$ in that case has interval of convergence $\vert(x-2)\rvert<2$ or $0<x<4$ which doesn't tell you the singular behavior of $y(x)$ near $x=0$. However a Frobenius series solution works well to let you see the behavior of the solution $y(x)$ in $0\le x<\infty$.