Find the general solution for the following non-homogeneous equation?

Question

$y^{(4)} - 2y''' + y'' = 2e^{3x} + x$

Attempt:

The characteristic equation is $r^4 - 2r^3 + r^2 = 0$.

==> $r^2 (r^2 - 2r + 1) = 0$

==> $r^2 (r - 1)^2 = 0 $

==> $r = 0, 0, 1, 1$.

So, $$y_h = (C_1 + C_2 x)e^{0x} + (C_3 + C_4 x)e^{1x}= C_1 + C_2 x + (C_3 + C_4 x)e^x.$$