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 $ℝ$, $u\ne 0.$

The paper describes the general form of an ordinary differential equation of the second order 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 $$f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz$$ is solved on $ℝ$ for $y\ne 0$, $v\ne 0.$

For linear differential and functional-differential equations of the $n$-th order criteria of equivalence with respect to the pointwise transformation are derived.

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.