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Solving Laguerre DEby translating it into an Euler equation

  1. May 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the indicial equation and all power series solutions around 0 of the form
    xr Ʃan xn for:
    x y'' -(4+x)y'+2y=0
    - apparently one of these solutions is a laguerre pilynomial

    2. Relevant equations
    the indicial equation is the roots of
    r(r-1) +p0r+q0
    where p0=lim(x->0)( x(-4-x)/x)=-4
    and q0=lim(x->0)( x^2 *2/x)=0
    Hence the indicial equation is:
    r^2-r - 4r =r(r-5)
    3. The attempt at a solution
    I have a solution for the root at r=5, but I'm not sure how to do it for r=0, which is the Laguerre one...?
     
    Last edited: May 4, 2012
  2. jcsd
  3. May 4, 2012 #2
    What is the difficulty here? Just plug your ansatz [itex] y = \sum_n a_n x^n [/itex] into the equation, and solve like usual. By googling "laguerre polynomial" you will see that you expect to get a solution which contains only 2+1 = 3 terms; you will probably find something like [itex] a_{n+1} \propto (n-2) a_n [/itex], meaning that for all n>2, an = 0.
     
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