Separation of variables for solutions of partial differential equation

[jamesb1]

Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?

 

[pasmith]

Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?

You don't need the boundary conditions to be homogeneous for the method to work, and conversely homogeneity of the boundary conditions does not guarantee that the method will work.

What you do need is for the problem to be linear, the boundary to consist of co-ordinate curves or surfaces, and for the equation to be separable in those co-ordinates.

Since the problem is linear, its solution can be written as a sum of solutions satisfying mostly homogeneous boundary conditions, which is what one needs in order to apply Sturm-Liouville theory.