Imagine a truck towing a car using a taught rope of constant length $l$. The trucks position at any time is $(x,y)$ and is known in parametric form:
$$x=f(t)$$ $$y=g(t)$$
The position of the car $(X,Y)$ is thus described by: $$X = x - l \cos\theta$$ $$Y = y - l \sin\theta$$
Since the car is forced to point towards the truck at all times, the following constraint applies: $$\tan(\theta) = \frac{\dot{Y}}{\dot{X}}$$
It should be possible to use this information to solve for the motion of the car assuming generic initial conditions. By use of algebra I get the following nonlinear coupled differential equation:
$$X = f(t) - \frac{l\dot{X}}{\sqrt{\dot{X}^2+\dot{Y}^2}}$$ $$Y = g(t) - \frac{l\dot{Y}}{\sqrt{\dot{X}^2+\dot{Y}^2}}$$
Can someone solve this? Perhaps by using a clever change of variables?
It may be better to write an equation for $\theta$. With $X = f(t) - l \cos(\theta)$ and $Y = g(t) - l \sin(\theta)$, I get $$ \dfrac{d\theta}{dt} = \dfrac{\sin(\theta) \dot{g}(t) - \cos(\theta) \dot{f}(t)}{l} $$ This will have closed-form solutions for some $\dot{f}$ and $\dot{g}$, but not arbitrary ones.