In bispherical coordinates, Laplace's equation becomes
(1)
|
Attempt separation of variables by plugging in the trial solution
(2)
|
then divide the result by to obtain
(3)
|
The function then separates with
(4)
|
giving solution
(5)
|
Plugging back in and dividing by gives
(6)
|
The function then separates with
(7)
|
giving solution
(8)
|
Plugging back in and multiplying by gives
(9)
|
so Laplace's equation is partially separable in bispherical coordinates. However, the Helmholtz differential equation cannot be separated in this manner.