Why does separation of variables produce solutions without arbitrary functions? The "general solution" (solutions without boundary conditions or initial conditions) to a PDE usually seems to include an arbitrary function of independent variables.  This make sense, because indefinite integration with respect to one variable produces an arbitrary function of the other variables.
However, when performing separation of variables, the "general solution" has arbitrary constants which depend on $n$ (often labeled $A$, $B$, and $\lambda$), but not arbitrary functions of the independent variables.
Why does the general solution to a PDE differ so fundamentally depending on the solution method used?
 A: When we separate variables, we assume that the solution can be written as the product of functions of the different variables. Then, when focusing on the equation in one of the variables only, we often find quantized solutions, due to the boundary conditions - a simple example is the Schrodinger equation in 2D for an infinite square well. The general solution will be a weighted sum (Fourier series) of these normal modes, with the weights being the arbitrary constants we index by n. If the spectrum of modes is continuous (for example, the Schrodinger equation for a particle in free space), we take a Fourier transform, with the weight being a function of whatever parameter is involved. If we then construct the full solution, each function (whether expanded in terms of its Fourier modes or not) will be multiplied by whatever function of the other variables was found when solving the relevant equations. Since we have solved the equation in all of the variables and already imposed the BCs in the process, those functions are determined.
The solutions appear different because the boundary (or initial) conditions are imposed at different stages of the process.
