Explicit solution, find x(t), y(t)

Question

I have a nonlinear system of equations:

$$\dot{x}=x(y-1)$$ $$\dot{y}=y^{2}(x-1)$$

And I wish to find x(t) and y(t) of this system. I might be missing a trick, but I just can't see to get it to work. The $xy$ and $y^{2}x$ are throwing me off. I am aware there are two equilibrium points at $(0,0)$ and $(1,1)$.

Any hints or what method to use are appreciated.

Answer 1

I do not think you can get explicit solutions for $x$ and $y$ as a function of $t$. But you can get the equations of the trajectories in the phase plane. From the equations we get $$ \frac{dy}{dx}=\frac{y^2(x-1)}{x(y-1)}\implies\frac{y-1}{y^2}\,dy=\frac{x-1}{x}\,dx. $$ Integrating $$ \log y+\frac1y=x-\log x+C, $$ or $$ x\,y\,e^{1/y-x}=C. $$

Answer 2

One direct approach -- from the second equation you have $$ x = 1 + y'y^{-2} $$ thus $$ x' = y''y^{-2} -2 y'y^{-3} $$ and you can now plug this into the first equation and solve for $y$ in a 1-variable ODE...

Important to keep in mind this assumes $y \ne 0$...