Hello. I am having some trouble with the following PDE:
Laplacian(u(x,y,z)) = u(x,y,z) * (-2*E/h^2)*(1 + (GM/(2(x^2+y^2+z^2)^(1/2))))^4
Where, G,M,E, and h are all constants.
The problem that I'm having is that there is a nonconstant factor of (x^2+y^2+z^2)^(-1/2) appearing on the RHS of this equation, making it non-trivial to solve. I tried separation of variables, and couldn't seem to get it to work.
Could someone please take a look?
This equation does not separate in rectangular coordinates, so you need to convert to spherical coordinates. In these coordinates, the equation becomes
Laplacian(u(r,theta,phi)) = u(r,theta,phi) * (-2*E/h^2)*(1 + (GM/(2r)))^4.
Writing out the Laplacian in spherical coordinates, this equation becomes
(1/r^2) d/dr(r^2 du/dr) + 1/(r^2 sin theta) d/dtheta(sin theta du/dtheta) + 1/(r^2 sin^2 theta) d^2 u/dphi^2
= u(r,theta,phi) * ...
We show how to solve a partial differential equation by first transforming the equation to spherical coordinates and then separating variables.