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Math Help - a second order diff. eq.

  1. #1
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    a second order diff. eq.

    Hi,

    Does anybody know if there's a method for solving:

    y''-cy+y^3=0 with the restriction \lim_{|x|\rightarrow\infty}y(x)=0 where c>0 is a constant.

    Thanks!
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  2. #2
    MHF Contributor
    Jester's Avatar
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    Re: a second order diff. eq.

    Well, here's maybe a starting place. Multplying the ODE by y' shows it can be integrated once. From your boundary condition (adding that maybe the derivative does the same) gives the constant of integration vanishes. Solving for y' gives (I've replaced the c by c^2/2)

     y' = \pm \dfrac{\sqrt{c^2y^2-y^4}}{2}

    or

     \dfrac{dy}{y\sqrt{c^2-y^2}} = \pm \dfrac{dx}{2}.

    You should be able to integrate this and see if your solution give the desired result at \pm \infty.
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