Find a Poincaré Map associated with the system \begin{align}\dot{r}&=(r-3)(r^2 -4) \\ \dot{\theta} &=4\end{align}
I have the following so far
\begin{align}\int\frac{1}{(r-3)(r^2-4)}dr &= \int dt \\ \therefore t + c &= \int\frac{1}{(r-3)(r^2-4)}dr\end{align} Now, using Partial fractions, we can rewrite the right-hand-side as
\begin{align} t + c &= \int\bigg(-\frac{1}{4 (r-2)}+\frac{1}{20 (r+2)}+\frac{1}{5 (r-3)}\bigg)dr \\ &=-\frac{1}{4}\ln|r-2| + \frac{1}{20}\ln|r+2| + \frac{1}{5}\ln|r-3| + k\end{align}
How can I reduce this into a a form $r(t) = \dots ~$?
I require this in order to continue with the rest of the question.