# Math Help - a second order diff. eq.

1. ## 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!

2. ## 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 $\pm \infty$.