Cross-ratios and first order ordinary differential equations

We investigate the usefulness of the cross-ratio in the study of first order odes of the form $y^\prime=P_ny^n+P_{n-1}y^{n-1}+\ldots+P_1y+P_0,\;\; (P_i)_{0\leqslant i\leqslant n}\in K,\;n\geqslant1$, where $K$ is a differential field of characteristic zero with algebraically closed field of constant