How to solve this system of ode Consider the system
$$
\begin{array}{r}
x^{\prime}=(x+y)\left(1-\sqrt{x^2+y^2}\right), \\
y^{\prime}=(y-x)\left(1-\sqrt{x^2+y^2}\right)
\end{array}
$$
of differential equations with $x, y \neq 0$.
So I have made the transformation to polar coordinates, and I got:
$$
\begin{array}{r}
r^{\prime}=r(1-r), \\
\theta^{\prime}=r(r-1)(cos2\theta)
\end{array}
$$
(hopefully, the calculations are not wrong).  My ode is a bit rusty so I am actually not sure how to solve this system or which method I should use.

$$\begin{array}{r}
r^{\prime}=r(1-r)\\
\theta^{\prime}=r(r-1)(\cos (2\theta))
\end{array}$$
First DE is separable
The second one is also separable:
$$\theta^{\prime}=r(r-1)\cos(2\theta)$$
$$\theta^{\prime}=-r'\cos(2\theta)$$
$$\dfrac {d\theta}{\cos (2\theta)}=-dr$$
