Jet space of a scalar ordinary differential equation

The $n$th jet space for a scalar function $u(x)$ has local coordinatesIt 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.

A scalar ordinary differential equation $\Delta (x,u,u^{\prime },\dots ,u^{(n)})=0$ defines a subset of the jet space of a scalar ordinary differential equation. Where $\Delta$ is continuously differentiable and $d\Delta \ne 0$, the implicit function theorem makes this subset a smooth local hypersurface, called the solution manifold. If $\partial \Delta /\partial u^{(n)}\ne 0$, it can locally be written as an explicit equation for the highest derivative.

The prolonged graph of every classical solution lies in this subset. At a singular point, however, the zero set need not be a manifold. For example $(u^{\prime }{)}^2-u^2=0$ is the union of the two sheets $u^{\prime }=u$ and $u^{\prime }=-u$ in first-order jet space of a scalar ordinary differential equation; they meet at $u=u^{\prime }=0$. Calling the whole zero set a smooth hypersurface would discard the required regularity condition.