Solution
= Solution
The <characteristic equation> factors as $(r+1)(r+2)=0$, giving $y_h=C_1e^{-x}+C_2e^{-2x}$. The <superposition principle> lets us find <particular solutions> for the two forcing terms separately. A trial $Ax+B$ gives $2A=1$ and $3A+2B=0$, so it is $x/2-3/4$. A trial $Ce^{3x}$ gives $(9+9+2)C=1$, so $C=1/20$. Therefore
$$
\boxed{y(x)=C_1e^{-x}+C_2e^{-2x}+\frac x2-\frac34+\frac1{20}e^{3x}}.
$$