Given the initial conditions $r=L$ at $\theta=0$, solve $r(t)$ and $\theta(t)$ for the following set of coupled differential equations. $$ \dot{r} = -u - v \sin \theta $$
$$ r \dot{\theta} = -v \cos \theta $$
This is in fact based on a physics problem: on the $xy$ plane, consider a person starting to run at speed $u$ to the $+y$ direction from the origin. A dog initially at $(L, 0)$ chases it at speed $u>v$ and always heads towards the person. Find the time needed for the dog to catch the person.
I have attempted to find an equation containing $r$ or $\theta$ only. By isolating $v$ and the trig functions, I modified the first equation to $\dot{r} + u = -v\sin \theta$. By squaring and adding the two equations, I arrive at
$$\dot{r}^2 + 2\dot{r}u + r^2 \dot{\theta}^2 + (u^2-v^2)=0$$
By differentiating the first and set it equal to the second, one finds $$\ddot{r} = r \dot{\theta}^2$$.
Substituting this yields $$\dot{r}^2 + 2\dot{r}u + r \ddot{r} + (u^2-v^2)=0$$
How can I go further? Or should I try an alternative approach?