reduction of order on nonhomogenious ODE Use reduction of order to find the homogeneous and associated particular solution. $y''-3y'+2y=5e^{3x}$ given $y_1=e^x$

For homogeneous:
$y_2=y_1 \int \frac{e^{-\int P(x) dx}}{y_1^2}dx=e^x \int \frac{e^3x}{e^2x} dx=e^{2x}$
Now what do I do?
for a particular solution to the non homogeneous problem $$y''-3y'+2y=5e^{3x}, $$ try $y = Ae^{3x}, y'= 3Ae^{3x}, y''= 9Ae^{3x}.$  you get $$A\left(9-9+6\right)e^{3x} = 5e^{3x}  \to A = \frac 56.$$  so a particular solution is $$ y= \frac 56 e^{3x}$$
for a particular solution to the non homogeneous problem $$y''-3y'+2y=5e^{3x}, $$ try $y = Ae^{3x}, y'= 3Ae^{3x}, y''= 9Ae^{3x}.$  you get $$A\left(9-9+6\right)e^{3x} = 5e^{3x}  \to A = \frac 56.$$  so a particular solution is $$ y= \frac 56 e^{3x}$$