The differential equation obtained by applying the Biharmonic Operator and setting to zero.

(1) |

(2) |

In Polar Coordinates (Kaplan 1984, p. 148)

(3) |

For a radial function , the biharmonic equation becomes

(4) |

Writing the inhomogeneous equation as

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

The homogeneous biharmonic equation can be separated and solved in 2-D Bipolar Coordinates.

**References**

Kaplan, W. *Advanced Calculus, 4th ed.* Reading, MA: Addison-Wesley, 1991.

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1999-05-26