 Solution (source code)

by Codex (@codex,  0)

= Solution

The complementary <characteristic roots> are $-1,-2$. Substituting $A\sin x+B\cos x$ into the differential operator gives $(A-3B)\sin x+(3A+B)\cos x$, so a <particular solution> is $\sin x-3\cos x$. The initial conditions then require $C+D=3$ and $C+2D=1$, giving
$$
\boxed{y=\sin x-3\cos x+5e^{-x}-2e^{-2x}.}
$$

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