If and then . Also, if R stays finite as then as , and so . Those are the boundary conditions for X.
Consider the heat equation in a radially symmetric sphere of radius unity:
with boundary conditions
Now, using separation of variables leads to the eigenvalue problem
Then using the change of variable this becomes
Now to find all the eigenvalues I need to know what the boundary conditions are, and from the info above, how do I determine the sets of boundary conditions to use to find the eigenvalues ??