 Solution

The extremal equation is $y^{\prime \prime }+\lambda y=0$. For $\lambda >0$, its general solution is $A\cos (\sqrt{\lambda }x)+B\sin (\sqrt{\lambda }x)$. The left endpoint forces $A=0$, while the right endpoint and $B\ne 0$ require $\sin (\sqrt{\lambda }\pi )=0$. ThereforeIn particular, each extremal has the requested Fourier sine series form $y={\sum }_{k\ge 1}{a}_k\sin (kx)$. More strongly, for an individual extremal only the coefficient at $k=n$ can be nonzero. An arbitrary combination of distinct modes is an admissible function, but does not solve this one-eigenvalue equation.