The paper describes the general form of an ordinary differential equation of the order $n+1$ $(n\ge 1)$ which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form \[ f\biggl (s, v, w_{11}v_{1}, \ldots , \sum _{j=1}^{n}w_{nj}v_{j}\biggr ) = \sum _{j=1}^{n}w_{n+1 j}v_{j} + w_{n+1 n+1}f(x, v, v_{1}, \ldots , v_{n}), \] where $ w_{ij} = a_{ij}(x_{1}, \ldots , x_{i-j+1}) $ are given functions, $ w_{n+1 1} = g(x, x_{1}, \ldots , x_{n})$, is solved on $\mathbb R.$.
We prove that the problem of finding all ${\mathcal {M} f_m}$-natural operators ${C\colon Q\rightsquigarrow QT^{r*}}$ lifting classical linear connections $\nabla $ on $m$-manifolds $M$ into classical linear connections $C_M(\nabla )$ on the $r$-th order cotangent bundle $T^{r*}M=J^r(M,\mathbb R )_0$ of $M$ can be reduced to the well known one of describing all $\mathcal {M} f_m$-natural operators $D\colon Q\rightsquigarrow \bigotimes ^pT\otimes \bigotimes ^qT^*$ sending classical linear connections $\nabla $ on $m$-manifolds $M$ into tensor fields $D_M(\nabla )$ of type $(p,q)$ on $M$.