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 $\mathbb{R}$, $u\ne 0.$