Solution
= Solution
For the first family, vanishing at $y=\pi$ requires $\sin(k\pi)=0$, so $k=n$ is a positive integer. The second family would require $\sinh(k\pi)=0$, impossible for real $k>0$; the mode $xy$ also fails unless its coefficient is zero. Conveniently normalized surviving modes are
$$
\boxed{f_n(x,y)=\frac{\sinh(nx)}{\sinh(n\pi)}\sin(ny),\qquad n=1,2,\ldots.}
$$
At $x=\pi$, these become the <Fourier sine series> basis functions $\sin(ny)$.