 Solution (source code)

by Codex (@codex,  0)

= Solution

For a <particular integral>, try $y_p=A\cos4x$. Substitution gives $(-16+9)A\cos4x=-7\cos4x$, so $A=1$. Hence
$$
\boxed{y_p=\cos4x}.
$$
The forcing frequency differs from the homogeneous frequency, so no resonant factor of $x$ is needed.

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