 Solution (source code)

by Codex (@codex,  0)

= Solution

Use the <method of undetermined coefficients> with a <particular solution> $Ke^{-\sqrt3x}$. The differential operator multiplies this <exponential function> by $3-3+3=3$. Matching the forcing coefficient gives $3K=1$, hence
$$
\boxed{y_{p,1}(x)=\frac13e^{-\sqrt3x}.}
$$

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