I have following ODE problem
$(x)\frac{d^2y}{dx^2}+\frac{dy}{dx}+y/x=0$
The power series solution of that ODE near x=1 is
$y_{k+2}=-[\frac{(2k+1)(k+1)y_{k+1}+(k^2+1)y_k}{(k+2)(k+1)}]$
I have to find Radius of Convergence.
As $k\to \infty$
$y_{k+2}=-[\frac{2y_{k+1}+y_k}{1}]$
$\frac{y_{k+2}}{y_{k+1}}=-2-\frac{y_{k}}{y_{k+1}}$
Let $l=\lim_{k\to \infty}\frac{y_{k+2}}{y_{k+1}}$ ....My Assumption Limit exist and equals to ROC
then $l=-2-1/l$
Solving this I get $|l|=1$
So ROC is 1
Is this solution Mathematically Look correct? Please give me a suggestion
Any Help will be appreciated