Geodesic
On transformations $z(t)=y(\varphi(t))$ of ordinary differential equations
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 509-518 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\mathbb R.$
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 $\mathbb R.$
Classification : 34A25, 34A30, 34A34, 39B22, 39B40
Keywords: ordinary differential equations; linear differential equations; transformations; functional equations 
@article{CMJ_2000_50_3_a4,
     author = {Tryhuk, V\'aclav},
     title = {On transformations $z(t)=y(\varphi(t))$ of ordinary differential equations},
     journal = {Czechoslovak Mathematical Journal},
     pages = {509--518},
     year = {2000},
     volume = {50},
     number = {3},
     mrnumber = {1777472},
     zbl = {1079.34505},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a4/}
}
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Tryhuk, Václav. On transformations $z(t)=y(\varphi(t))$ of ordinary differential equations. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 509-518. http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a4/