Suppose we have this PDE problem $\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$ $\psi(0,t)=\psi(L,t)=0$ It represents the vibrations of a string tightly stretched between two points. The standard technique for the solution is separation of variables $\psi(x,t)=T(t)y(x)$, giving the equations $y''=\frac{1}{c^2}\lambda y$ $T''=\lambda T$.
Every text I consulted assumes then that $\lambda<0$ and goes on. They say that the physics of the problem requires the solution to be a combination of sines and cosines. But is there a more rigorous mathematical way to see this? Are we losing possible solutions? What could happen if $\lambda>0$?