schwarzschild | time WebGL visualization of a Schwarzschild Black Hole | Code Inspection library

It seems like you are looking at the spacial coordinates in an invariant plane of the geodesic equations of an object moving in Schwarzschild gravity.

One can use many different methods, which preserve as much of the underlying geometric structure of the model as possible, like symplectic geometric integrators or perturbation theory. However, I suspect that the default method for solve_ivp() is 4th order Runge-Kutta, which is a fairly good, direct and simple method.

Warnng: your initial condition for Y equals Schwarzschild's radius, so these equations may fail or require special treatment (especially the time component of the equations, which you have not included here!) It may be that you have to switch to different coordinates, that remove the singularity at the even horizon. Moreover, the solutions may not be periodic curves, but quasi-periodic, so they may not close up nicely.

For a quick and dirty treatment, but possibly a fairly accurate one, I would differentiate the first equation