Given a map of a product of two manifolds into a third one, one can define its jets of separated orders $r$ and $s$. We study the functor $J$ of separated $(r;s)$-jets. We determine all natural transformations of $J$ into itself and we characterize the canonical exchange $J\to J^{s;r}$ from the naturality point of view.

We study the problem of the non-existence of natural transformations $J^r{J}^{s}Y\to J^s{J}^{r}Y$ of iterated jet functors depending on some geometric object on the base of Y.

The paper describes the general form of an ordinary differential equation of an order $n+1$ $(n\ge 1)$ which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $$f\left(s,w_{00}{v}_0,...,\sum _{j=0}^n{w}_{nj}{v}_j\right)=\sum _{j=0}^n{w}_{n+1j}{v}_j+w_{n+1n+1}f(x,v,v_1,...,v_n),$$ where $w_{n+10}=h(s,x,x_1,u,u_1,...,u_n)$, $w_{n+11}=g(s,x,x_1,...,x_n,u,u_1,...,u_n)$ and $w_{ij}=a_{ij}(x_1,...,x_{i-j+1},u,u_1,...,u_{i-j})$ for the given functions $a_{ij}$ is solved on $ℝ$, $u\ne 0.$

We study geometrical properties of natural transformations $T^{A}T*\to T*T^A$ depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations $T^{A}T*\to T*T^A$ can be described in a uniform way by means of a simple geometrical construction.

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 $$f\left(s,v,w_{11}{v}_1,...,\sum _{j=1}^n{w}_{nj}{v}_j\right)=\sum _{j=1}^n{w}_{n+1j}{v}_j+w_{n+1n+1}f(x,v,v_1,...,v_n),$$ where $w_{ij}=a_{ij}(x_1,...,x_{i-j+1})$ are given functions, $w_{n+11}=g(x,x_1,...,x_n)$, is solved on $ℝ.$