How to solve this Cauchy's problem?

Question

How to solve this Cauchy's problem?

$\frac{dx_1}{dt}= \frac{x_1Q(t)}{2 \pi (x_1^2+x_2^2)}, $

$\frac{dx_2}{dt}= \frac{x_2Q(t)}{2 \pi (x_1^2+x_2^2)},$

$x_i(0)=\xi_i$

There $Q(t)>0$ is some "good" function

Answer 1

Let $r:=\sqrt{x_1^2+x_2^2}$ so$$r\frac{dr}{dt}=\frac12\frac{d}{dt}(x_1^2+x_2^2)=x_1\frac{dx_1}{dt}+x_2\frac{dx_2}{dt}=\frac{Q(t)}{2\pi},$$i.e. $x_1^2+x_2^2-\xi_1^2-\xi_2^2=\frac{1}{\pi}\int_0^tQ\left(t^\prime\right)dt^\prime$. Note also that $\frac{d}{dt}\ln\frac{x_1}{x_2}=0$, so $x_1\propto x_2$.