Solution
= Solution
On the <jet space of a scalar ordinary differential equation>, the $n$th <prolongation of a vector field> is
$$
\operatorname{pr}^{(n)}V
=V+\sum_{j=1}^n\eta^{(j)}\partial_{u^{(j)}},
\qquad
\eta^{(j)}=D_x\eta^{(j-1)}-u^{(j)}D_x\xi,
\qquad \eta^{(0)}=\eta,
$$
where $D_x$ is the <total derivative operator>. In particular,
$$
\boxed{\operatorname{pr}^{(1)}V
=\xi\partial_x+\eta\partial_u+
\left[\eta_x+(\eta_u-\xi_x)u'-\xi_u(u')^2\right]\partial_{u'}.}
$$