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On the jet space of a scalar ordinary differential equation, the $n$th prolongation of a vector field iswhere $D_x$ is the total derivative operator. In particular,

Write $u_j=u^{(j)}$. The transformation ${\psi }_s:(x,u)\mapsto (\tilde{x},\tilde{u})$ induces a transformation of the $n$th jet space of a scalar ordinary differential equation by differentiating $\tilde{u}$ with respect to $\tilde{x}$ through order $n$. The infinitesimal generator of this induced action is the prolongation of a vector fieldTo determine its coefficients, put ${\eta }_0=\eta$ and use the total derivative operatorDifferentiating transformed derivatives byand then differentiating with respect to the group parameter at $s=0$ gives the recursionThis also proves the displayed form of the prolongation by mathematical induction on the derivative order.

Write $\xi =V_1$, $\eta =\varphi$, and $u_j=d^{j}u/d{x}^j$. On the jet space of a scalar ordinary differential equation, definewhere the total derivative operator isThe order-$N$ prolongation of a vector field is

In the coordinates $(x,u)$, the given action of the special orthogonal group isDifferentiating at $s=0$ gives the infinitesimal generator of a Lie point symmetryso $\xi =u$ and $\eta =-x$. The recursive formula givesandTherefore the second prolongation of the rotation generator for plane graphs is

For ${\Delta }_0=u_{xx}$,which vanishes when ${\Delta }_0=0$. Forwe obtainwhich likewise vanishes on the equation. This is the infinitesimal invariance criterion for both equations.

Geometrically, $u_{xx}=0$ describes straight lines. The signed curvature of a plane graph isso the second equation says $\kappa =1$ and describes consistently oriented arcs of unit circles. Rotation preserves straight lines, circles, and signed curvature, explaining both invariances.

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.