this is my first post so I'll try to be brief.
I am working on finding a problem such as this:
given the ODE: $$(x^2-1)y''+x^2y' + cotx\cdot y = 0$$ "Find all the singular points of the following ODEs and determine whether each one is regular or irregular. If the singular point is regular, determine the indicial equation (i.e. the characteristic equation for the corresponding Cauchy-Euler equation) and deter- mine a lower bound for the radius of convergence of the Frobenius series."
My solution thus far:
Divide the ODE by the leading term's variable coefficient to get into the form: $$y'' + \frac{x^2}{(x^2-1)} y' + \frac{cotx}{(x^2-1)} y = 0$$ We can see that the 2nd and third variable coefficients will be undefined for x = 1,-1, or k*pi, where k is an integer including 0.
Now, taking x = 1 as the first singular point to check, lets see if it is regular or irregular.
$$\alpha_0 = \lim_{x \to 1} (x-1) \frac{x^2}{(x^2-1)} = \lim_{x \to 1}\frac{x^2}{(x+1)} = \frac{1}{2}$$
$$\beta_0 = \lim_{x \to 1} (x-1)^2 \frac{cotx}{(x^2-1)} = \lim_{x \to 1}\frac{cos(x)(x-1)}{sin(x)(x+1)} = 0$$
Since both are finite, this point is a regular singular point and we can try the Frobenius Series.
Substituting into the corespondent Cauchy-Euler Equation $$(x-1)^2y'' + \alpha(x-1)y'+\beta y = 0 $$
The indicial equation, with corresponding alpha and beta terms with the Frobenius Series as a solution: $$r(r-1) + \frac{r}{2} = 0$$
Thus, r = 0, 1/2
I am not sure how I can use this as an indicator of the lower bound radius of convergence of the Frobenius Series: As far as I'm concerned, the r values indicate the two linearly independant exponents in the Frobenius Series: $$\sum_{n=0}^{\infty}a_n(x-x_0)^{n+r}$$
EDIT: I realized i needed to apply the theorem of convergence, however, I still don't understand how to apply it as the point of expansion for the theorem of convergence is the singular point which I am looking at.
Any help would be appreciated.
Credit to Lutz Lehmann in the comment:
Finding the distance from the nearest singular point is one approach, another approach would be using the ratio test for the recursion equation given once you evaluate the sums.