= Solution
The <overdamped filament bending equation> is linear, so sinusoidal forcing has a sinusoidal steady response at the same frequency. Write $h=h_0\operatorname{Re}[e^{i\omega t}F(\eta)]$ with $\eta=x/\ell$. Substitution gives
$$
i\omega\zeta F=-\frac{A}{\ell^4}F^{(4)}.
$$
Using the <elastohydrodynamic penetration length> from the previous part reduces this <ordinary differential equation> to
$$
\boxed{F^{(4)}+iF=0.}
$$
The boundary conditions become $F(0)=1$, $F''(0)=0$, and decay of $F$ and its derivatives at infinity. The derivative condition expresses zero <bending moment> at the driven end, rather than a clamped slope. This <separation of variables> constructs the periodic response; an unspecified initial shape can add a transient solution of the homogeneous evolution problem.
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