I think my question is pretty simple but I'm struggling to understand how to find indicial equations and would really appreciate some help.
I have this equation:
$$4xy''(x) +2y'(x) + y(x) = 0$$
I want to find the indicial equation.
I started by finding $P(x)$ and $Q(x)$ which i believe are $1/2x$ and $1/4x$ respectively.
I believe $ x=0$ is a singular point (i might be wrong).
Where i get stuck is here, I know the next step involves limits, it might be along the lines of something like: $lim(x-0)P(x)$ but I'm really not sure.
If someone could help out and guide me through what I would need to do next that would help a lot. I know this might be quite simple but I'm quite new to this concept and don't really understand it.
Thanks in advance
For finding indicial roots put $y=x^m$ and compare the coefficient of the smallest power of $x$ to zero. $$4xy''(x) +2y'(x) + y(x) = 0 \implies 4m(m-1)x^{m-1}+2m x^{m-1}+x^m=0$$ $$\implies x^{m-1}(4m^2-2m)+x^m.$$ $m=0,1/2$ are the indicial roots.