TITLE: What is the motivation behind a product solution?
QUESTION [11 upvotes]: Let's consider the simple differential equation:
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$
And let's assume we have some regular homogeneous boundary conditions like:
$$ u(a, y) = 0$$ 
$$ u(L, y) = 0$$ 
$$ u'(x, y) = 0$$ 
(I just made these up at the top of my head so feel free to improvise if they don't work with what you want to say.)
My question is: What is the intuition/motivation needed in taking a product solution of the form: $$\Psi = X(x)Y(y)$$ From what I understand, we also assume that the BCs apply to $\Psi$ too.
From what  I've been told, there is no proof of this, it's just a reasonable trial that seems to work well in a lot of cases.

REPLY [3 votes]: With the proper selection of coordinates, any differential equation that is solvable in integrals (sequentially) separates.  (This is the meat and potatoes of the Lie symmetry techniques in solving differential equations.)  If your system has a lot of "obvious" symmetry, then the separation of variables corresponding to the obvious symmetry is a good thing to try first.  
Solving in a box?  Try $X(x,t)\cdot Y(y,t) \cdot Z(z,t)$.
Solving in a cylinder?  Try $R(r,t) \cdot \Theta(\theta,t) \cdot Z(z,t)$.
Et c...