How to solve the coupled system of ODES $\dot{x}=y^2$, $\dot{y}=x$?

Question

How to solve the following system of coupled ODEs?

$$ \begin{aligned} \dot{x} &= y^2 \\ \dot{y} &= x \end{aligned} $$

I can see how to solve for their trajectories as $dy/dx$ is of course separable but

1. Is there a transformation I can use to solve this?

2. If so, what is the motivation for the transformation?