First prolongation
= First prolongation
{title2=$\operatorname{pr}^{(1)}V$}
For $V=\xi(x,u)\partial_x+\eta(x,u)\partial_u$, the first prolongation is
$$
\operatorname{pr}^{(1)}V
=V+\left[\eta_x+(\eta_u-\xi_x)u'-\xi_u(u')^2\right]\partial_{u'}.
$$
= First prolongation
{title2=$\operatorname{pr}^{(1)}V$}
For $V=\xi(x,u)\partial_x+\eta(x,u)\partial_u$, the first prolongation is
$$
\operatorname{pr}^{(1)}V
=V+\left[\eta_x+(\eta_u-\xi_x)u'-\xi_u(u')^2\right]\partial_{u'}.
$$