On transformations $z(t)=y(\varphi(t))$ of ordinary differential equations

Tryhuk, Václav

Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 509-518.

Résumé

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 $\mathbb R.$

Détail

Classification : 34A25, 34A30, 34A34, 39B22, 39B40
Keywords: ordinary differential equations; linear differential equations; transformations; functional equations