An interesting differential equations problem Suppose we have four ants, initially at rest, at the four corners of a square centered at the origin. They start walking clockwise, each ant walking directly toward the one in front of him. Suppose also that each ant walks with unit velocity, derive a differential equation that describes the trajectories.
Thought:
Here is the situation of the problem

After some time $t$, the ants are now at points $E,F,G,H$. If we denote by $\mathbf{r(t)}$ the position of ant at $A$, we know $\mathbf{r(0)} = (1,1)$ and after some $t$, at point $E$, we have $\mathbf{r(t)} = (x(t),y(t)) = E $. We are given that
$$ \frac{ y - 1 }{x - 1 } = 1 $$
Also, using the arclength formula, we know the path of ant $A$ is
$$ \int\limits_0^t \sqrt{ (x')^2 + (y')^2 } dt $$
Am I on the right track?

In general, the differential equation that each ant (indexed $i,$ and cyclic so that $n + 1$ is the same as $1$) follows is
$\frac{d\bf{r}_i}{dt} = v \frac{\bf{r}_{i+1} - \bf{r}_i}{||\bf{r}_{i+1} - \bf{r}_{i}||}$
where $v$ is the ant's speed and $||\cdot||$ denotes the vector length (2-norm).
As the other answer mentions, for this special case where the ants are arranged in a square, it is possible to set up simpler equations to find the trajectories.
