 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 areAt $x=\pi$, these become the Fourier sine series basis functions $\sin (ny)$.