Differential equation in polar coordinates I have the following system:
$\frac{dx}{dt} = 3x + y - x(x^2+y^2)$
$\frac{dy}{dt} = -x +3y -y(x^2+y^2)$
Converting this to polar coordinates gives us:
$\frac{dr}{dt} = r(3-r^2)$
$\frac{d\theta}{dt} = -1$
This gives us a solution $\theta(t) = -t + \theta_0$. What would the solution for $r(t)$ be though?

You have
$$\frac{1}{r (\sqrt{3}-r)(\sqrt{3}+r)} dr = dt.$$ Use partial fractions on the left hand side and integrate.
