Were Analytical Solutions Used For Galileo and Cassini?
The fact that models will always faile to predict what they were not intended to does not mean they can't reasonably predict what they are intended to. Even Newton's laws don't yield analytical solutions when applied to a large number of bodies. I'm pretty sure it was computer models that gave us all those precise close passes of outer planet Moons. Just like a failed model crashed the polar lander. The fault in that model was readily identified once people actually checked.
Regards,
Mark