Hi there, for the following equation that is in the phase plane (x,y),

$\displaystyle d^{2}x/dt^{2}-4x+x^{3}=0$ <--- sorry, there was a typo originally, now correct

where $\displaystyle dx/dt=y$

I need to prove that the solutions of the coupled, nonlinear system are given by

$\displaystyle y^{2}=1/2(C+4-x^{2})(C-4+x^{2})$

where C is a constant. So what I have done so far is:

$\displaystyle d^{2}x/dt^{2}-4x+x^{3}=0$

So the equivalent system is:

$\displaystyle F(x,y) = dx/dt=y$ (as stated in the question)

$\displaystyle G(x,y) = dy/dt-4x+x^{3}=0 => dy/dt=4x-x^{3}$

So from this, I tried letting

$\displaystyle dy/dx=(dy/dt)/(dx/dt)=G(x,y)/F(x,y)$

This upon solving this, I got $\displaystyle y^{2}=4x^{2}-(x^{4})/4+2C$

which seems close, but I am not sure this is the right method or not