We use Milman's method to find a particular solution:
$$L\left[A\frac{x^m}{m!}e^{rx}\right]=Ae^{rx}\left(Q(r)\frac{x^m}{m!}+Q'(r)\frac{x^{m-1}}{(m-1)!}+Q''(r)\frac{x^{m-2}}{2!(m-2)!}+... \right)$$
In this case
$Q=r^2-3r+2, Q'=2r-3, Q''=2$. We want to have $r=3$. So we evaluate that $Q(3)=2, Q'(3)=3, Q''(3)=2$.
Also, let $m=0$. Then $L(Ae^{3x})=2Ae^{3x}$. This implies that $A=1$.
In conclusion, solution is $Y_p=e^{3x}$.
For general solution to homogeneous equation, solve $r^2-3r+2=0$ yielding $r_1=2$, $r_2=1$.
So, $Y_{gen.hom}=c_1e^{2x}+c_2e^{x}$
General solution for the whole system is $Y_G=Y_{gen.hom}+Y_p=c_1e^{2x}+c_2e^{x}+e^{3x}$.
Condition $Z(0)=1$ yields that $c_1+c_2+1=1$. Condition $Z'(0)=0$ yields that $2c_1+c_2+3=0$. Solving these gives $c_1=-3, c_2=3$.
In conclusion we get that: $y=-3e^{2x}+3e^x+e^{3x}$.
