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Math Help - y'' + 2/x y' = cy

  1. #1
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    y'' + 2/x y' = cy

    Hello!

    I have this second order differential equation which is causing me some trouble:

    y'' + 2/x y' = c y

    where c is a constant. I've got no clear idea of where to start solving it, the 2/x term is the main source of my head ache here.
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  2. #2
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    Re: y'' + 2/x y' = cy

    First start by writing it clearly. Do you mean y''+ 2/(xy')= cy or y''+ (2/x)y'= cy? Since you refer to "2/x", I guess you mean the latter. That is a linear second order differential equation with variable coefficients which has a regular singularity at x= 0. If you multiply through by x you get xy''+ 2y'- cy= 0. The simplest thing to do is to look for a power series of the form \sum_{i= 0}a_n(x- b)^n where b is NOT 0. Or look for a series solution of the form \sum_{n= 0}^\infty a_nx^{n- c} where c is a positive number.
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  3. #3
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    Re: y'' + 2/x y' = cy

    Thank you! Yes, my apologies, I meant the latter. So xy''+ 2y'- cxy= 0 would be the result from the multiplication, if I guess that there is an x missing from the last term, and if so does it change the power series approach?

    And a little follow-up question: Is it always possible to find a power series-solution to a differential equation?
    Last edited by Mikael; March 8th 2014 at 05:53 AM.
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  4. #4
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    Re: y'' + 2/x y' = cy

    No need for power series in this case.
    More simply, let y=z/x which leads to a second order ODE easy to solve for z(x)
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  5. #5
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    Re: y'' + 2/x y' = cy

    My power series did not really work out well so I'll get back to that later with more questions, but if I could solve it as an ODE then it would make my day but I got stuck on that one as well.

    I tried calculating y' and y'' for y = z(x)/x but this gives pretty heavy expressions to substitute and becomes even tougher, so I suppose I'm working from the completely wrong direction.
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  6. #6
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    Re: y'' + 2/x y' = cy

    the substitution $y=z/x$ is the correct one and will lead you to a linear 2nd order diff eq w/constant coefficients which is easily solved.

    Keep plugging away.
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  7. #7
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    Re: y'' + 2/x y' = cy

    Thank you guys, I think I got quite close right now. Wolfram alpha gets a slightly different answer than I get, but I'll try to put it all here tomorrow to see if someone agrees/disagrees with me.

    And btw: Is it possible to get a solution through a power series for this one (and really any differential equation, or are there restrictions)? I'd, ultimately, at some point have to practice on doing those and this would be a good time if it's possible.
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