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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- Sep 19th 2011, 11:45 AMTheAppa 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! - Sep 19th 2011, 02:40 PMJesterRe: 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 $\displaystyle \pm \infty$.