= Solution
For the forcing $8e^{-2x}$, the <method of undetermined coefficients> gives the particular term $2e^{-2x}$. The term $e^{-x}$ is resonant with a homogeneous solution, so use $axe^{-x}$ for its forcing; substitution gives $a=1$.
Thus $y=C_1e^{2x}+C_2e^{-x}+2e^{-2x}+xe^{-x}$. The two initial conditions reduce to $C_1+C_2=3$ and $2C_1-C_2=0$, so $C_1=1$, $C_2=2$. The solution of the <initial value problem> is
$$
\boxed{y(x)=e^{2x}+(x+2)e^{-x}+2e^{-2x}.}
$$
The extra factor $x$ in the resonant particular solution is essential.
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