I am trying to solve $\ rR''+2R'+rR\lambda=0\ $, given $R(1)=0$ and $R'(0)=0$.
I let $S(r)=rR$ which gave the ODE $$S''(r)+\lambda S(r)=0.$$ I know wish to solve this problem, but I'm having trouble converting the boundary condition $R'(0)=0$ into a condition on $S(r)$. If $R(1)=0\implies S(1)=0$. But how does $$R'(r)=\frac{rS'(r)-S(r)}{r^2}\Bigg|_{r=0}$$ relate to a boundary condition for $S(r)$?