In two-dimensional Cartesian coordinates, attempt separation of variables by writing

(1)

then the Helmholtz differential equation becomes

(2)

Dividing both sides by  gives

(3)

This leads to the two coupled ordinary differential equations with a separation constant ,

			(4)
			(5)

where  and  could be interchanged depending on the boundary conditions. These have solutions

			(6)
			(7)
			(8)

The general solution is then

(9)

In three-dimensional Cartesian coordinates, attempt separation of variables by writing

(10)

then the Helmholtz differential equation becomes

(11)

Dividing both sides by  gives

(12)

This leads to the three coupled differential equations

			(13)
			(14)
			(15)

where , , and  could be permuted depending on boundary conditions. The general solution is therefore

(16)

REFERENCES:

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 501-502, 513-514 and 656, 1953.