When solving a partial differential equation (PDE) using separation of variables, it is assumed that the solution to the PDE is the product of functions of each variable. For examle, given:
$$ \psi (x,y,z)=X(x)Y(y)Z(z)$$
In what type of PDEs is this valid?
How can we know that no solutions are being 'ignored' by making this assumption?
Are there any cases in which we can be sure that all the solutions can be written as the product of functions?