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Separation of Variables for PDEs

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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?

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Solution Preview

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) * ...

Solution Summary

We show how to solve a partial differential equation by first transforming the equation to spherical coordinates and then separating variables.