Suppose we have a system in polar coordinates:
$\dot r = -r$ and $\dot \theta = \frac{1}{\ln{r}}$, we are asked to solve for $r(t)$ and $\theta(t)$ explicitly, so I just integrated both equations so that $r = Ae^{-t}$ and $\theta(t) = \frac{t}{\ln r} + C$ where $A,C$ are constants.
How would I write this system in terms of x and y?