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Thread: 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:

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

    Thanks!
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  2. #2
    MHF Contributor
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    Re: a second order diff. eq.

    Well, here's maybe a starting place. Multplying the ODE by $\displaystyle y'$ 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 $\displaystyle y'$ gives (I've replaced the $\displaystyle c$ by $\displaystyle c^2/2$)

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

    or

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

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