Here is what I have so far $y_g=C_1x^{1/2}+C_2x^{1/2}lnx$
We write
$y_p=U_1(x)x^{1/2}+U_2(x)x^{1/2}lnx$ $y_p'=x^{1/2}U_1'+x^{1/2}lnxU_2'+1/2x^{-1/2}U_1+(1/2x^{-1/2}lnx+x^{-1/2})U_2$
We choose to impose that $x^{1/2}U_1'+x^{1/2}lnxU_2'=0$
Then $y_p''=-1/4x^{-3/2}U_1+1/2x^{-1/2}U_1'+(1/2x^{-3/2}-1/4x^{-3/2}lnx)U_2+(1/2x^{-1/2}lnx+x^{-1/2})U_2'$
Subbing into the ODE and multiplying out we have
$-x^{7/2}U_1+2x^{3/2}U_1'+(2x^{1/2}-x^{1/2}lnx)U_2+2x^{3/2}lnx+x^{3/2}U_2'+U_1x^1/2+U_2x^{1/2}lnx$
Now i'm not sure if my equations are correct but i'm pretty sure I've made a mistake because I think that somehow I show be able to take the final equation and the one that was imposed to 0 and take them away from each other but if they are right I have no idea how to do this.