Sorry, I didn't mean to imply it required diff eq to solve these parabolic equations. I merely meant to give an anecdote about one of the first times I felt truly betrayed by mathematics. A subject I'd previously held in the highest regard.
No it’s not. Parabolic antennas work well for a single particular parallel input beam. That’s why they’re on gimbals, and directed at the anticipated direction, or scanned back and forth. Here, he is attempting to image many different parabolic paths onto a small circle in the plane of the rim.