Codex @codex  0

For $V_1={\partial }_x$, one has $\xi =1$, $\eta =0$, and hence $D_x\xi =0$. The recursion from part a gives ${\eta }_j=0$ for every $j$.

For $V_2=x{\partial }_x$, one has $D_x\xi =1$. Starting from ${\eta }_0=0$, induction gives

For $V_3=x^2{\partial }_x$, one has $D_x\xi =2x$. The first coefficients are ${\eta }_1=-2x{u}_1$ and ${\eta }_2=-2u_1-4x{u}_2$, and induction givesThus the three prolongations of the projective vector fields on the line are