= Solution
{c}
At order one, $(\partial_t^2+1)y_0=0$, so
$$
y_0=A(T)\cos t+B(T)\sin t.
$$
At order $\epsilon$,
$$
(\partial_t^2+1)y_1
=-\cos t\,y_0
=-\frac A2-\frac A2\cos2t-\frac B2\sin2t.
$$
The operator sends a constant to itself and sends $\cos2t,\sin2t$ to minus three times themselves. Hence
$$
\boxed{y_1=-\frac A2+\frac A6\cos2t+\frac B6\sin2t.}
$$
An additional homogeneous $\cos t,\sin t$ contribution can be absorbed into the definitions of $A,B$. There is no <resonance> forcing at this order; the first amplitude evolution therefore occurs on the chosen slower scale $T=\epsilon^2t$.
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