It is a well known (in the physics world) that an orbit of 3 or more objects cannot be solved perfectly. We call it the "1, 2 and you're done" rule. So, as soon as you have something like the Sun, Earth and Moon, there is no equation that tells you where the objects will be as a function of time. Throw in all the planets of the Solar System and it gets even worse.

But, since we have powerful computers, this is ok, we can still simulate it to really high accuracy. And in fact, when you start to do simulations, the math actually gets easier instead of harder- you just have to do something easy a whole bunch of times (so much of studying physics and engineering is doing complex math so that you can approximate things without having to do them in a simulation which can take a lot of processing power).

So, to determine orbits we do something like this. First, take all of the positions, velocities of the planets you're looking at, and calculate all of the gravitational forces acting on them right now. This is easy, you're not saying "how is the gravitational force changing?" you're just saying "what is it now?" so you can use Newton's law of -G*M*m/r² , and do that for all the combinations. That allows you to calculate an instantaneous acceleration for every object, and then by using standard motion formulas, you predict where the object will be in some "small" time step in the future (and what "small" is is determined by what you're working with and how accurate you want it to be. For something like planets, you might say "small" is an hour, and for something like a satellite orbiting the Earth you might say "small" is a second). So, you simply say p_dt = p_now + v_now*dt + 1/2*a_now*dt² (position after some time "dt" is equal to the position now plus the velocity now times that "small" change plus one half of the acceleration now time that "small" change of time squared).

Then, you simply repeat the process, over and over again, until you're as far into the future as you want to me. The further you predict, the less accurate you will be, and the smaller of step size you make the more accurate you will be.

I must say, this is a simplification of how real orbit propagators work- but it does capture the basics (and it is how simple things like Kerbal Space Program works). Real orbit propagators track too many objects to do a full simulation like this- it would take too much processing power. So instead they propagate the orbit assuming that only the biggest mass nearby matters (like say, the Sun) and then all of the other masses (the other planets) act as small perturbations to that predicted motion. The nice thing about this is that motion of one planet around one sun can be predicted easily, and then just small changes to that motion are added up to see how it changes thing. But this is a much more complicated process than described above, and the basics of it are the same.