Let $A$ be the set of solutions. I think that if we can show these solutions are dense in $L^2(\Omega)$ then this would mean that we have found every solution.
If we show that $A$ vanishes nowhere and is separable then $A$ is dense in $L^2(\Omega)$ by Stone-Weierstrass, but I am not sure how to show this for an arbitrary set of solutions. Is it true that these solutions are all orthogonal since we have separated variables?