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Legendre's equation

  1. Dec 26, 2008 #1
    1. The problem statement, all variables and given/known data

    Obtain the recurrene relation between the coefficient ar in the series solution

    y= (between r=0 and [tex]\infty[/tex]) [tex]\Sigma[/tex] arxr

    to (1-x2)y''-2xy'+k(k+1)y=0
    Deduce that if k is a positive integer, then ak+2=0, so that the equation possesses a solution which is a polynomial of degree k.

    2. Relevant equations

    (there is more to this question....but i think ill try getting through this bit first!)


    3. The attempt at a solution

    I have managed to get the correct recurrance relation

    (n+2)(n+1)an+2-an(n(n+1)-k(k+1))=0

    But i have no idea what to do now (the show if k is a positive integer) etc :-(
     
  2. jcsd
  3. Dec 26, 2008 #2

    HallsofIvy

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    So
    [tex]a_{n+2}= a_n\frac{n(n+1)- k(k+1)}{(n+1)(n+2)}[/tex]
    Assume that a0 and a1[/sup] are given. What is a2, a3, a4 in terms of a0 and a1. Can you guess a form for the general an form? Can you then prove it by, say, induction?
     
  4. Dec 26, 2008 #3
    i am not too sure what induction is.....(sorry!) guessing a form for an - do you is this not given by the recurrance relation?
     
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