In 2-D Cartesian Coordinates, attempt Separation of Variables by writing

(1) |

(2) |

(3) |

(4) | |||

(5) |

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

(6) | |||

(7) |

The general solution is then

(8) |

In 3-D Cartesian Coordinates, attempt Separation of Variables by writing

(9) |

(10) |

(11) |

(12) | |||

(13) | |||

(14) |

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

(15) |

**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.

© 1996-9

1999-05-25