I've been given the following problem, namely I have to study whether the solution of this system of differential equations $\begin{cases}\dot x=\cos(xy)x^3\\ \dot y=\cos(xy)y^3\end{cases}$ is defined on the whole of $\mathbb R$ as $(x(0),y(0))$ vary in $\mathbb R^2$.
I've begun my analysis considering the constant solutions. Due to the symmetry I may WLOG assume that $\dot x=0$ because if it were $\dot y=0$, then we would get the same conclusions. So if $\dot x=0$ then we may have $\cos (xy)=0$ which means $xy=(2k+1)\pi/2$, but, plugging this into the second, even $\dot y=0$, hence $y$ constant and then the only requirement needed would be $x(0)y(0)=(2k+1)\pi/2,\;k\in\mathbb Z$. If instead $x=0$, we would get $\dot y=y^3$, and then, eventually $\frac{1}{y(t)^4}=\frac{1}{y(0)^4}-4t.$ In this case $y$ cannot be defined on the whole of $\mathbb R$. This conclude my first part of the solution.
Then i passed to consider non constant solutions, and clearly I divided out the first equation by the second and I got $\dot x/x^3=\dot y/y^3$, which led me to $\frac{1}{x(t)^4}=\frac{1}{y(t)^4}-\left(\frac{1}{y(0)^4}-\frac{1}{x(0)^4}\right),$ but I couldn't say anything more from here, and thus I would like to see what else can be inferred from this discussion or to see different approaches to the problem in order to finish it clearly. Thank you very much for your kindness.
Regards.