Right, with 3 balls you can't just do an impulse calculation like you can with 2 (too many unknowns, too few equations).

The author is simulating the collision as an event which happens gradually over an (admittedly short) period of time, using a differential equation, instead of something that happens all at once. That's why he has to talk about the elasticity model he's using, F ∝ (2-d)^1.5, and how the results vary when you change it.

Could you get more equations by symmetry?

Yes … in the symmetric case. That might give you enough to solve the first one-against-two collision. But the subsequent collisions in the pool break problem won't be symmetric (the velocity of the "incoming" ball is not perpendicular to the axis joining the centers of the other two).

Hmm. How does the elasticity model provide the additional constraint needed to give an unique solution?

It provides a way to apportion the impulse between the two simultaneous collisions. It behaves as a differential equation because the instantaneous force in each collision depends (via the elasticity model) on the instantaneous velocities of the balls, which in turn depend on how much momentum has already been transferred.