Finding solutions to a non homogeneous differential equation knowing the solutions of its homogenous counterpart 
Let $f(x)$ and $xf(x)$ be the particular solutions of a differential
  equation $$y''+R(x)y'+S(x)y=0$$ Find the solution of the differential
  equation $$y''+R(x)y'+S(x)y=f(x)$$ in terms of $f(x)$.

I was trying to find a solution if the form $(ax^2+bx+c)f(x)$. Now as $(bx+c)f(x)$ is a solution to the homogeneous equation, I just need to assume that $ax^2f(x)$ is a particular solution of the non homogeneous equation, and find the suitable value of $a$. But I am getting stuck. Please help.

Once you found two linearly independent solution of the homogeneous equation  say $y_1$ and $y_2$, the particular solution is $$ y_p= u_1y_1+u_2 y_2$$Substitute in your inhomogeneous equation and solve for $u_1$ and $u_2.$ You may study the method of varion of constant for specific examples. 
