The second-order Ordinary Differential Equation

(1) |

(2) |

(3) | |||

(4) | |||

(5) |

Plugging in,

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

Therefore,

(13) | |||

(14) | |||

(15) |

so the Even solution is

(16) |

Similarly, the Odd solution is

(17) |

If is an Even Integer, the series reduces to a Polynomial of degree with only Even
Powers of and the series diverges. If is an Odd Integer, the series reduces
to a Polynomial of degree with only Odd Powers of and the series diverges. The
general solution for an Integer is given by the Legendre Polynomials

(18) |

(19) |

The *associated* Legendre differential equation is

(20) |

(21) |

(22) |

(23) |

(24) |

and

(25) |

(26) |

Plugging (22) into (26) and the result back into (21) gives

(27) |

(28) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 332, 1972.

© 1996-9

1999-05-26