How do we show that $(0,0)$ is the only equilibrium point?

Question

For the following system, $$ x'=x-y-x^3 $$ $$ y'=x+y-y^3 $$ I get solve that $(0,0)$ is the equilibrium. But how do we show that $(0,0)$ is the only equilibrium point?

That means we need to solve $$ x-y-x^3=0 $$ $$ x+y-y^3=0 $$

Answer 1

Hint: From the second equation, we have $$x = y^3-y$$

Substitute that into the first equation, expand and get a ninth-order equation in $y$ only.

It only has a single real root $y = 0$, and then use that in the above to find $x$.

The other eight roots are imaginary.