[aoanla]

N-body problems can't be analytically solved. However, you can still compute integrals into the future (with some acceptable error), you just need to step through all the intermediate states along the way

In the case of the solar system, yes, it helps that the Sun is much more massive than everything else (and then Jupiter is 4 times more massive than Saturn, the next biggest) - you can go a long way to a "reasonable" solution by starting with the 2-body solution if only the Sun affected each planet, and then adding in the perturbation caused by Jupiter and Saturn. (In fact, that's how we predicted the existence of Neptune, by noticing that there were extra perturbations on Uranus beyond those, and hence another massive planet must exist, far enough away from the sun to only significantly affect Uranus).