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Math Help - 3-Term Recurrence Relation Series Soultion

  1. #1
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    3-Term Recurrence Relation Series Soultion

    Hello everyone!

    I'm studying for power series solution for ODEs and I came over this example:
    y''-(1+x)\,y=0. And by assuming a power series centered at the ordinary point x_0=0 we're bound to get two equalities:
    c_2=\frac{1}{2}\,c_0 and c_{k+2}=\frac{c_k+c_{k-1}}{(k+1)(k+2)} for k=1,2,3...

    I went right ahead to continue working the solutions out but I accidently looked at the solution and it shows to sets of calculations: one assuming c_0=0 \mbox{ and} \ c_1 \mbox{ different than } \ 0 and another one supposing the opposite.

    Can someone explain why is this assumption valid?

    Thanks!!
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  2. #2
    MHF Contributor chisigma's Avatar
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    The assumption is valid because the second order ODE is linear and its general solution has the form...

    y(x) = c_{1}\cdot \varphi_{1} (x) + c_{2}\cdot \varphi_{2} (x) (1)

    ... where \varphi_{1} (x) and \varphi_{2} (x) are two particular independent solutions, c_{1} and c_{2} two 'arbitrary constants'...

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by rebghb View Post
    Hello everyone!

    I'm studying for power series solution for ODEs and I came over this example:
    y''-(1+x)\,y=0. And by assuming a power series centered at the ordinary point x_0=0 we're bound to get two equalities:
    c_2=\frac{1}{2}\,c_0 and c_{k+2}=\frac{c_k+c_{k-1}}{(k+1)(k+2)} for k=1,2,3...

    I went right ahead to continue working the solutions out but I accidently looked at the solution and it shows to sets of calculations: one assuming c_0=0 \mbox{ and} \ c_1 \mbox{ different than } \ 0 and another one supposing the opposite.

    Can someone explain why is this assumption valid?

    Thanks!!
    Any initial conditions given?

    I believe that the authors wanted to avoid the messy algebra and wanted to isolate each solution separately (Chisigma hinted at that).
    Last edited by Jester; April 25th 2010 at 06:28 AM. Reason: Added more info
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  4. #4
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    No it's not an IVP or anything, it's just the concept I am after.
    Thanks anyways!
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