Solving second order nonlinear ODE given boundary condition at infinity

Question

I am trying to solve the following differential equation $$\frac{d^2 u}{dx^2} = - \frac{d V}{du} \; \; , \;\; where \;\; \; V = \frac{1}{2}u^2 - \frac{1}{4}u^4 $$ And the given boundary conditions are $u(-\infty)=1$ and $u(\infty)=-1$. The solution that I should get it $$u(x) = \mp \tanh \left( \frac{x}{\sqrt{2}} \right) ,$$ but how do I get there?

Answer 1

$$ u'' = u'\dfrac{d}{du}u' $$ Change of variables $u'\to p$ we get $$ \dfrac{d}{du}p^2 = -2\dfrac{dV}{du} $$ You have now reduced the order to proceed.

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