Solution
= Solution
The <characteristic equation of a constant-coefficient differential equation> here is $r^2-3r+2=(r-1)(r-2)=0$. Its <homogeneous solution> is $C_1e^x+C_2e^{2x}$. Because the forcing $e^x$ is already a homogeneous mode, the <method of undetermined coefficients> requires $y_p=Axe^x$ rather than $Ae^x$.
Here $y_p'=A(x+1)e^x$ and $y_p''=A(x+2)e^x$, so $y_p''-3y_p'+2y_p=-Ae^x$. Thus $A=-1$, and the <general solution> is
$$
\boxed{y(x)=C_1e^x+C_2e^{2x}-xe^x.}
$$