= Solution
{c}
Use the fast time $T_0=t$ and slow time $T=\epsilon^2t$. A possible intermediate scale $\epsilon t$ has no leading modulation: the absence of first-order <resonance> imposes independence of that scale. Write
$$
U_0=A(T)e^{iT_0}+\overline A(T)e^{-iT_0}.
$$
At first order choose the nonresonant particular solution
$$
U_1=2|A|^2-\tfrac13A^2e^{2iT_0}-\tfrac13\overline A^{,2}e^{-2iT_0},
$$
with any needed fundamental homogeneous correction assigned consistently when matching the initial data. At second order,
$$
(D_0^2+1)U_2=2U_0U_1-2D_0D_TU_0.
$$
The coefficient of $e^{iT_0}$ on the right is $(10/3)|A|^2A-2iA_T$. A bounded oscillatory correction requires this <resonance> coefficient, and its conjugate, to vanish. The <quadratic oscillator phase-modulation condition> is therefore
$$
\boxed{A_T=-\frac{5i}{3}|A|^2A.}
$$
It resums the secular phase accumulation into a slow modulation instead of leaving it as a growing term in the correction.
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