= Solution
The extremal equation is $y''+\lambda y=0$. 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 $B\ne0$ require $\sin(\sqrt\lambda\pi)=0$. Therefore
$$
\boxed{\lambda=n^2,\qquad y(x)=B\sin(nx)\quad(n\in\mathbb N).}
$$
In particular, each extremal has the requested <Fourier sine series> form $y=\sum_{k\geq1}a_k\sin(kx)$. 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.
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