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?
Thanks!
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Solution Summary
We show how to solve a partial differential equation by first transforming the equation to spherical coordinates and then separating variables.
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) * ...
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