Use Green's function to reduce to quadratures the solution of \begin{equation} xy''+y'+xy=R(x), ~~~~~~~~~~ u(0) \;\text{finite}, u(b)=0, \end{equation} on $0\leq x\leq b,$ where $b=y_{0,1}=0.89377$ is the first zero of the $Y_0$ Bessel function and $R$ is bounded on $[0,b].$ You may use the result that the Wronskian of the ${J_0}$ and $Y_0$ Bessel function is $W(j_0,Y_0;x)=J_0(x)Y_0'(x) -J_0'(x)Y_0(x)=\dfrac{2}{(\pi x)}$ [There is a subtle difference to the standard case here, due to $x=0$ being a singular point. Is it not necessary that one of your linearly independent solutions of the homogeneous equation vanish at $x=0?$ Instead, you require only that one solution be bounded there.]
My work so far:
\begin{eqnarray} xy''+y'+xy&=&R(x) \nonumber\\ xy''+y'+xy&=&0\nonumber\\ \text{let} ~y&=&x^n\implies y'=nx^{n-1} \nonumber\\ \implies y''&=&n(n-1)x^{n-2}\nonumber\\ n(n+1)+n+1&=&0\nonumber\\ n^2-n+n+1&=&0\nonumber\\ n^2+1&=&0\nonumber\\ n^2&=&-1. \end{eqnarray} How to continue?