We have the linear differential equation
$$9x^2y''+9xy'-(1+x)y=0.$$
I want to determine and characterize the singular points of the equation in $\mathbb{R}$.
I have done he following.
A Sturm-Liouville problem has the following form.
$$-(p(x)\phi'(x))'+q(x) \phi(x)=\lambda \phi(x), x \in [a,b].$$
If $p$ gets zero at some point, then the Sturm Liouville problem is singular.
$$(x^2 y')'=2xy'+x^2 y''$$
$$9x^2 y''=9(x^2 y')'-18 xy'$$
So, $9(x^2y')'-9xy'-(1+x)y=0$.
Can this be writtenin the form of the Sturm-Liouville problem?
Instead take your original equation and divide by $x$. You get:
$$9xy’’+9y’-(1+1/x)y=0.$$
Now notice the first two terms are $(9xy’)’$. Can you finish from here?