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Math Help - URGENT! power series solution of differential equation

  1. #1
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    URGENT! power series solution of differential equation

      xy''+(1-x)y'+\lambda y=0
    find the power series solution about x=0 with y(0)=1 and show:

    y is a polynomial iff lambda is a non-negative integer n.
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  2. #2
    MHF Contributor chisigma's Avatar
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    We start supposing that the second order linear ODE...

    x\cdot y^{''} + (1-x)\cdot y^{'} + \lambda\cdot y =0 , y(0)=1 (1)

    ... has an analytic solution of the form...

    y(x)= \sum_{n=0}^{\infty} a_{n}\cdot x^{n} (2)

    If we substitute the expression (2) directly in (1) we obtain...

    \sum_{n=0}^{\infty} \{(\lambda - n)\cdot a_{n} + (n+1)^{2} \cdot a_{n+1}\}\cdot x^{n} =0 (3)

    ... which immediately gives us...

    a_{n+1} = \frac{n-\lambda}{(n+1)^2}\cdot a_{n} (4)

    If we set a_{0}=1 the (4) allow us to derive all the a_{n}. It is evident from (4) that if \lambda= k with k an integer, all the a_{n} with n>k vanish, so that y(*) is a polynomial of degree k...

    Kind regards

    \chi \sigma
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