I've just encountered the following differential equation in a design problem
$\dot{l}\cos \theta - \dot{\theta}l \sin \theta = A[\dfrac{1}{\sqrt{r}}-\dfrac{1}{\sqrt{r+l\sin \theta}}] \quad (*)$.
Here $A$ and $r$ are constant parameters. Regarding the meaning of the parameters, we know that $r \gg l$, thereby $r \approx r+l\sin \theta $. Thus, the RHS can be neglected which leaves us with the differential equation
$\dot{l}\cos \theta - \dot{\theta}l \sin \theta = 0$,
which is easy to be solved. ($\dfrac{\dot{l}}{l} = \dot{\theta}\tan \theta$, so $\ln{l} = \ln \dfrac{1}{|\cos \theta|}$, and finally $l = \dfrac{1}{|\cos \theta|}$).
If one relaxes the $r \approx r+l\sin \theta $ assumption, is there any closed solution to the equation $(*)$?