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How can this be solved if it is assumed that the satellite does not affect the motion of the Earth and the Moon?
In principle, it could be solved just by solving Newton equation for the case of potential being a sum of two gravitational potentials. Not sure it is a graduate-level question though.
At short times, where the position of the Earth and the Moon can be considered fixed, this is called the restricted 3-body problem, a very classical question exactly solved by Euler 250 years ago. See https://en.wikipedia.org/wiki/Restricted_3_body_problem
For longer times, one solves it stepwise by using the short time solution, updating the position of Earth and Moon after each time step, e.g., using ephemeris. https://en.wikipedia.org/wiki/Ephemeris
Without further information it is hard to tell if this is really a graduate-level question or just an investigation of a curious enthusiast...
Voting to close. 500+ rep users please vote on the closevote in the link if you (dis)agree.
Although not graduate level I find this question interesting and I learned something I did not know. I won't vote to close although strictly speaking, indeed, not graduate level.
@conformal_gk: Closed questions can still be viewed by clicking on the 'Closed Question' category on the side bar (towards the button). They can also be found by search on title key words. They are just no longer on the main menu.
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