= Solution
The <characteristic equation> of the homogeneous <ordinary differential equation> is $r^2-2r+5=0$, with roots $1\pm2i$. Therefore $y_h=e^x(C\cos2x+D\sin2x)$. For the forcing term, try the <particular solution> $y_p=e^{-x}(ax+b)$. Substitution gives $8a=1$, $8b-4a=0$, so $a=1/8$, $b=1/16$. The two initial conditions give $C=-1/16$ and $D=0$. Consequently
$$
\boxed{y(x)=\frac{(2x+1)e^{-x}-e^x\cos2x}{16}.}
$$
Both the value and the first derivative vanish at zero; direct differentiation verifies the inhomogeneous equation.
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