Solution
= Solution
With gravity omitted, the initial data contain only the first fixed-end <normal mode>, so write
$$
y(x,t)=q(t)\sin\frac{\pi x}{l}.
$$
Then
$$
q''+2kq'+\left(\frac{\pi c}{l}\right)^2q=0,
\qquad q(0)=0,\quad q'(0)=A.
$$
Because $k=\pi c/l$, this oscillator is <critical damping>[critically damped], and
$$
q(t)=Ate^{-kt}.
$$
Therefore
$$
\boxed{y(x,t)=Ate^{-kt}\sin\frac{\pi x}{l}}.
$$