The geodesic equations is the system of equations which often takes the form
$\frac{d}{dt}(Eu^{'}+Fv^{'})=\frac{1}{2}(E_{u}(u^{'})^{2}+F_{u}u^{'}v^{'} +G_{u}(v^{'})^{2})$
$\frac{d}{dt}(Fu^{'}+Gv^{'})=\frac{1}{2}(E_{v}(u^{'})^{2}+F_{v}u^{'}v^{'} +G_{v}(v^{'})^{2})$
It is not clear from the context whether these have variable coefficients or constant.
In general, the coefficients are not constant. Please see this question for an example.