Jet space of a scalar ordinary differential equation
= Jet space of a scalar ordinary differential equation
The $n$th jet space for a scalar function $u(x)$ has local coordinates
$$
(x,u,u_x,u_{xx},\ldots,u^{(n)}).
$$
It records the value of a function and its derivatives through order $n$ at one point, allowing an order-$n$ <ordinary differential equation> to be treated as a hypersurface and a point transformation to act through its <prolongation of a vector field>.