# Deriving logarithmic spiral equation from square corners

This is an interesting problem but I haven't been able to work it out (from Bender and Orszag, prob: 1.27) - any insight/assistance would be appreciated:

Four caterpillars, initially at rest at the four corners of a square centered at the origin, start walking clockwise, each caterpillar walking directly toward the one in front of it. If each caterpillar walks with unit velocity, show that the trajectories satisfy the differential equation

$$\frac{dy}{dx}=\frac{y-x}{y+x}$$

This is the equation for a logarithmic spiral in $$xy$$-coordinates. And I can see how the movement slowly traces points on rotating squares which get smaller and smaller in size, giving a logarithim spiral, but I cannot find a way to derive the equation from the given geometry in terms of dynamic parameters like $$dx, dy, dt, ...$$