Partial differential equations ( variable separable) There are lots of way to solve P.D.E , but I want to know that mostly in physics to solve P.D.E like Schrödinger equation we mainly use variable separable method. 
So can anyone tell that what is limitations for using variable separable method? 
And what makes variable separable so powerful to use in solving the P.D.E . 
plz help..
 A: An important limitation is that it works only for regions whose boundaries are level sets of the coordinate system you are using. In Cartesian coordinates, you are restricted to boxes. In Cylindrical coordinates, you're restricted to cylinders, cones, etc. The coordinate systems should be orthogonal, meaning that the coordinate curves intersect orthogonally; if this isn't true, mixed derivative terms creep into the transformed Laplacian, which is typically a show-stopper for separation of variables. When you put these restrictions together, there are no more than a couple of dozen viable coordinate systems. In addition to this, equations such as Schrodinger's equation where there is a potential are usually a problem unless the potential does not depend on more than a single variable. These are the types of restrictions you run into. But the technique was good enough to allow for explicit solutions of some very important cases. Without the invention of separation of variables, Science and Mathematics would have been severely hindered. Rigorous Mathematics, including integration, linear spaces, topology, and a host of other topics came about directly because of trying to justify the separation of variables solution methods and associated expansions.
What makes separation of variables so power is the reduction to ODEs, and the possibility of explicit solutions.
