I started studying the geodesic equation: ∂2xμ/∂s2 = - Γμab(∂xa/∂s)(∂xb/∂s) where the term s is proper time according to the wiki(https://en.wikipedia.org/wiki/Geodesics_in_general_relativity). The 2nd derivative on the left side of the equation is the acceleration in the xμ direction. Now my question is: When I solve the geodesic equation, what exactly am I trying to solve for? Am I solving for s (or rather am I supposed to plug in information into the equation in order to derive s)? If not s, then what exactly am I solving for? If you already have your Christoffel symbols derived, then you should have already long had your xμ functions derived (which is why I don't think you are solving for xμ). To be honest, this equation seems more to me like the Euler - Lagrange equations in the sense that, the main purpose doesn't seem to be to solve the equation, but more so to plug in known information in order to derive equations of motion.