I want to solve the geodesic equation on $\mathbb{CP}^n$ with the Fubini Study metric $$ g_{ij}=\frac{\left(1+{\mid z\mid}^2\right)\delta_{ij}-\bar{z}_iz_j}{\left(1+{\mid z\mid}^2\right)^2} $$ I have worked out the Christoffel symbols for the above metric. They are as follows $$ \Gamma^{k}_{ij}=-\frac{\delta^{k}_{j}\bar{z}_{i}+\delta^{k}_{i}\bar{z}_{j}}{\left(1+{\mid z\mid}^2\right)} $$ Thus the affinely parametrized geodesic equation becomes $$ \frac{d^2z^k(t)}{dt^2}-\frac{2\bar{z}_{j}(t)}{\left(1+{\mid z(t)\mid}^2\right)}\frac{dz^k(t)}{dt}\frac{dz^j(t)}{dt}=0 $$ I want to solve this equation for the case of $n=1,2$. I have never solved an ODE with complex variables.
Any suggestion or help is appreciated.