Two things going on here.
First. The hawk is always pointing at the pigeon. If the pigeon is located at and the hawk is travelling along (think of and as the path of the hawk), then the slope of the tangent to the curve will equal the slope of the line that connects the two locations (of the hawk and pigeon). So
Next we need to bring in the speed of the hawk. Since the hawk is travelling along this curve , the
Now eliminate in this system ((1) and (2)) to get a single second order DE for .
Hope this helps.