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Solution to Legendre equation in trig form

  1. Apr 19, 2012 #1
    hey guys,

    I've been trying to solve this question,

    http://img515.imageshack.us/img515/2583/asfj.jpg [Broken]

    so the general solution would be

    y(cos(theta)) = C Pn(cos(theta)) + D Qn(cos(theta)) right?

    and since n = 2 in this case

    y(cos(theta)) = C P_2 (cos(theta)) + D Q_2 (cos(theta))

    and 0<= theta < 2Pi

    But when theta = 0, cos(theta) = 1 and Q_2 is undefined, so D = 0,

    so y(cos(theta)) = C P_2 (cos(theta)) = (C/2) (cos^2(theta) - 1)

    but

    y(cos(0))=y(1) = C P_2(1) = C = ?

    So would the 2pi periodic solution be

    y(cos(theta)) = (C/2) (cos^2(theta) - 1) ?

    Thanks
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Apr 21, 2012 #2

    vela

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    You should have ##P_2(x) = \frac{1}{2}(3x^2 - 1)##, but otherwise your work looks fine.
     
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