Non linear first order ODE on $\mathbb{R}^2$

Question

Consider the system of ODEs $$\left\{ \begin{matrix} x' = y^2, & x(0) = x_0\\ y' = x^2, & y(0) = y_0 \end{matrix} \right. $$

where $x, y :\mathbb{R} \to \mathbb{C}.$ I wonder whether there is a "known formula" for the solution of such systems.

Thank you for any information

Answer 1

This system considered as

$$ \dot x = g(y)\\ \dot y = g(x) $$

can be handled by dividing both equations giving

$$ \frac{dy}{dx} = \frac{g(x)}{g(y)} $$

which is separable

Attached a stream plot for

$$ \dot x = y^2\\ \dot y = x^2 $$

with a typical solution in red

$$ y^3-x^3 = C_0 $$

Answer 2

You can try this $$\frac {dx}{dy}=\frac {y^2}{x^2}$$ $$\implies x^3=y^3+K$$