I was trying to visualize the meaning of a certain properties of some multivariable functions.
For simplicity, let's consider a two variables function, whose graph can be easily visualized: f(x,y).
In electromagnetic problems, there are many situations in which these functions show a separate dependence on x and y:
f(x,y) = X(x)*Y(y)
Is this property associated to a particular graphical feature/symmetry/other property?
Assuming you are talking about separable partial differential equations, when performing Seperation of Variables, in general the actual "final function" (the infinite superposition of all separable solutions) is not necessarily a function that can take the form ${f(x,y) = X(x)Y(y)}$. The steps go as follows
(1) Assume a separable solution exists
(2) Apply the standard argument to reduce to a system of ODEs
(3) With nice homogeneous boundary conditions, solve this new Sturm Liouville eigenvalue problem. This will give us a potentially infinite number of separable solutions (note these small separable solutions will also not in general satisfy our wanted initial conditions).
(4) Superpose all these solutions. That is, take a linear combination of all of them. By the (what I will assume) linearity of the PDE, this new function will be a solution too. Note at this stage the solution no longer necessarily satisfies ${f(x,y)=X(x)Y(y)}$
(5) Find the coefficients of the infinite linear combination that will satisfy our initial conditions. Usually done with a bunch of integral trickery (and by using the orthogonality of the eigenvalue problem)
(6) ???
(7) Profit
So really, at it's core it works because of Sturm Liouville theory. It really is just a Mathematical convenience rather than anything physical I think, but someone correct me if I am wrong