Fine a solution bounded near $x=0$ of the following ODE $$x^2y''+xy'+( \lambda ^2x^2-1)y=0$$
my attempt :
this is Bessel's equation so let $u=\lambda x$ then $y(x)=y(\frac{u}{\lambda})$
Also $y'(x)=\lambda Y'(u)$ and $y''(x)=\lambda^2Y''(u)$
then the give equation reduced to $u^2Y''(u)+uY'(u)+(\lambda^2-1)Y(u)=0$
then the general solution is $Y(u)=AJ_1(u)+BY_1(u)=AJ_1(\lambda x)+BY_1(\lambda x)=y(x)$
But what a solution bounded near $x=0$ ?