How can one find and prove the general solution to the equation $\dfrac{\partial f(x,t)}{\partial t} =c^2i\dfrac{\partial^2f(x,t)}{\partial x^2}$ ?
I can find the solutions $Ae^{ikx-E_kt}$, so I expect linear sombinations of this to solve the equation, but can, and if so why, every solution be written as $\int a(k)e^{ikx-iE_kt}~dk$?
I think its two questions,
a) (Why) can every 'basis' solution be found by separation of variables?
b) (Why) is the general solution an integral and not a sum of all LI. solutions?