plotting – Plot3D Equation of motion of a particle in a surface

I’m new in this forum. This semester I’ve been trying to learn how to use Mathematica for my Classical Mechanics course, but I’m still a novice concerning this software. In particular today I was doing a problem about a water drop on a surface $z=x^2-y^2$ and I came with Lagrangian Mechanics to the following equations:

$$(1+4x^2)ddot{x}-4yddot{y}x+4xdot{x}-4dot{y}^2x-2gx=0 $$
$$(1+4y^2)ddot{y}-4xddot{x}y+4ydot{y}-4dot{x}^2y+2gx=0 $$

where $g=9,8$ and arbitrary initial conditions. I tried a lot of codes, but I always had mistakes. Please, can you help me how to Plot3D this trajectory? Thanks in advance.