Consider the following system of first order nonlinear autonomous ODEs (derivatives are taken with respect to $t$): $$ \begin{cases} \dot{x} = -2xy^2+1 \\ \dot{y} = -2x^2y+1 \\ x(0) = x_0,\;y(0)=y_0 \end{cases} $$ Prove that
- if initial condition satisfies $x_0=y_0$, then $x(t)=y(t)$ for all $t\geq 0$ and $x(t)\to 1/\sqrt[3]{2}$ as $t\to\infty$.
- if $x_0\ne y_0$, then the solution is unbounded, i.e., $|x(t)|+|y(t)|\to\infty$ as $t\to\infty$.
For (1), my intuition is that if $x_0=y_0$, then $\dot{x}=-2x^3+1$ and $\dot{y}=-2y^3+1$, i.e., $x$ and $y$ are decoupled. However, I don't know how to argue it rigorously. If this is indeed true, then it is easy because this system has only one equilibrium, i.e., $x^*=1/\sqrt[3]{2}$, and if $x(t)>x^*$, $\dot{x}<0$, so $x(t)$ is decreasing. Similarly, if $x(t)<x^*$, $\dot{x}>0$, so $x(t)$ is increasing.
For (2), I have no idea how to approach. Any hint is welcome.