Laplaces Equation in Spherical Coordinates

Laplace s equation in spherical polar coordinates can be written as 

This can hold tme for all values of r, 9, and 0 only if each of the two parts of this PDE equals a constant. Let the first part equal — k and the second equal k. The ODE for r becomes 

Returning to the spherical harmonics, these evidently satisfy a PDE in two variables  

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