# Indicial equation of $4xy''+2y'+y=0$?

We consider $$4xy''+2y'+y=0$$ and search for a power series solution $$y=\sum_{r=0}^{\infty} a_rx^{r+m},$$ where we assume $a_0\neq 0$. By differentiating and substituting into the differential equation we get $$\sum_{r=0}^{\infty} 2a_r(r+m)(2r+2m-1)x^{r+m-1}+\sum_{r=0}^{\infty} a_rx^{r+m}=0.$$ The lowest power of $x$ is $x^m$ whose coefficient gives the indicial equation $$2m(2m-1)a_0+a_0=0.$$

But shouldn't the roots of the indicial equation be $m=0$ or $m=1/2$?

I would appreciate any help or suggestion on this.

Thank you.

The lowest power appearing in the equation (as you have it), is actually $x^{m-1}$, so the indicial equation (coming from $r=0$) would be:
$$2a_0 m (2m-1)=0$$
so indeed your roots are $m=0$ and $m=1/2$.