dx/dt = x(x^3 - 2y^3)
dy/dt = y(2x^3 - y^3)
Please help!
If y = xv then your ODE becomes
$\displaystyle \dfrac{dv}{dx} = \dfrac{v(1+v^3) }{1-2v^3} \dfrac{1}{x}$
which separate to
$\displaystyle \dfrac{1-2v^3}{v(1+v^3)} dv = \dfrac{dx}{x}$
or
$\displaystyle \left(\dfrac{1}{v} - \dfrac{3v^2}{v^3+1} \right) dv = \dfrac{dx}{x}$ (this easily integrates!)