Specific Solution to Differential Equation

Solve $x{(x-1)}^{2}g{}^{\u2033}\left(x\right)+(x-1){g}^{\prime}\left(x\right)-xg\left(x\right)=\frac{x-1}{x},{\textstyle \phantom{\rule{2em}{0ex}}}x\ge 1.$

I already know the general solutions to this as

$g\left(x\right)={c}_{1}P\left(x\right)+{c}_{2}Q\left(x\right)+{g}_{s}\left(x\right).$

Here, $P\left(x\right)=x{P}_{\alpha}^{2}(2x-1),{\textstyle \phantom{\rule{2em}{0ex}}}Q\left(x\right)=x{Q}_{\alpha}^{2}(2x-1),{\textstyle \phantom{\rule{2em}{0ex}}}\alpha =\frac{\sqrt{5}-1}{2}$

P and Q are the associated Legendre functions and ${g}_{s}\left(x\right)$ is a special solution to this equation. I also know that the special solution can be determined by an integral

${g}_{s}\left(x\right)=2Q\left(x\right)\int \frac{P\left(x\right)}{{x}^{3}}dx-2P\left(x\right)\int \frac{Q\left(x\right)}{{x}^{3}}dx$.

Can anyone help me find a particular solution to this equation?