In two-dimensional Cartesian coordinates, attempt separation of variables by writing
(1)
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then the Helmholtz differential equation becomes
(2)
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Dividing both sides by gives
(3)
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This leads to the two coupled ordinary differential equations with a separation constant ,
(4)
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(5)
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where and could be interchanged depending on the boundary conditions. These have solutions
(6)
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(7)
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(8)
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The general solution is then
(9)
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In three-dimensional Cartesian coordinates, attempt separation of variables by writing
(10)
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then the Helmholtz differential equation becomes
(11)
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Dividing both sides by gives
(12)
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This leads to the three coupled differential equations
(13)
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(14)
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(15)
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where , , and could be permuted depending on boundary conditions. The general solution is therefore
(16)
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