= Solution
The first mode remains the solution from part (b). The applied force excites only the third mode, so
$$
y(x,t)=Ate^{-kt}\sin\frac{\pi x}{l}
+q_3(t)\sin\frac{3\pi x}{l},
$$
where, using $\pi c/l=k$,
$$
q_3''+2kq_3'+9k^2q_3=\alpha\cos kt,
\qquad q_3(0)=q_3'(0)=0.
$$
A particular solution is
$$
q_{3,p}(t)=\frac{\alpha}{34k^2}(4\cos kt+\sin kt).
$$
Adding the decaying complementary solution and imposing the initial conditions gives
$$
\boxed{
q_3(t)=\frac{\alpha}{34k^2}
\left[
4\cos kt+\sin kt
-e^{-kt}\left(
4\cos(2\sqrt2kt)
+\frac5{\sqrt2}\sin(2\sqrt2kt)
\right)
\right]}.
$$
As $t\to\infty$, all transients decay but the driven third-mode oscillation remains. The string approaches a periodic <steady-state response>, rather than coming to rest.
With gravity omitted, its mechanical energy is
$$
E=\frac{\mu}{2}\int_0^l
\left(y_t^2+c^2y_x^2\right)dx.
$$
Writing $s=\alpha/(34k^2)$ and letting $t\to\infty$ gives
$$
\boxed{
E_\infty(t)
=\frac{\mu l\alpha^2}{4624k^2}
\left[
25\sin^2kt+145\cos^2kt
+64\sin kt\cos kt
\right]}.
$$
The limiting energy is periodic because the external force continually supplies the energy dissipated by drag.
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