Regular and irregular points of second order ODE

Question

I want to find the regular and irregular singular points of this ODE

$$x \sin (x) y'' + 3y' + xy = 0$$

What can I do?

Answer 1

To start of, I think you can see that when $x=0, x=n\pi$ and $x=\infty$, these might be singularity points. Divide the whole equation by $xsin(x)$ and see if the corresponding coefficients are analytic or not.

What I meant by that is to check if $\frac{1}{sin(x)}$ and $\frac{x}{sin(x)}$ are analytic about $x=0$. This is because recall you can write the most prototypical ODE in the form of $y"+\frac{p(x)}{x-x_0}y'+\frac{q(x)}{(x-x_0)^2}=0$ and then one has to check if $p$ and $q$ are analytic about $x_0$.

Do the same for $n\pi$ where $n$ is non-zero.

Then do the same for the point at infinity via the change of variables, $x=\frac{1}{z}$.