A friend of mine posed this question to me:
A boat with a maximum speed of $5$ meters per second crosses a river that is flowing at $3$ meters per second, and is fifteen meters wide. The boat always faces its target $(0,15)$. Does the boat reach its target? If so, how long does it take? What is the path of the boat?
If the boat is at a point $(x,y)$, then the angle $\theta$ to the target is $\theta(x,y) = \arctan\left(\frac{15 - y}{x}\right)$ and the velocity is $v(\theta) = (-3 + 5 \cos \theta) \hat{\imath} + (5 \sin \theta) \hat{\jmath}$.
I couldn't find parametric equations, so I wrote a small program to solve it for me. The program steps through small intervals of time, moves the boat by $v\,\mathrm dt$, and recalculates the velocity. The program halts when the boat has arrived at the target.
The graph of the path of the boat generated by this program should approximate the actual graph. The program found that the time taken is about $4.7$ seconds.
How can I find parametric equations for the path of the boat?