Frobenius & Special Functions · Challenge problem

The equation $x^2y'' - 3xy' + 3y = x^3$ is resonant: $x^3$ matches one of the homogeneous solutions ($y=x$ and $y=x^3$ solve the indicial equation $r^2-4r+3=0$), so the particular solution needs a factor of $\ln x$. For the solution with $y(1)=0$, $y'(1)=0$, find $y(e)$. Give an exact answer.