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

• Volume: 50, Issue: 3, page 509-518
• ISSN: 0011-4642

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## Abstract

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The paper describes the general form of an ordinary differential equation of the order $n+1$$\left(n\ge 1\right)$ 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 $f\left(s,v,{w}_{11}{v}_{1},...,\sum _{j=1}^{n}{w}_{nj}{v}_{j}\right)=\sum _{j=1}^{n}{w}_{n+1j}{v}_{j}+{w}_{n+1n+1}f\left(x,v,{v}_{1},...,{v}_{n}\right),$ where ${w}_{ij}={a}_{ij}\left({x}_{1},...,{x}_{i-j+1}\right)$ are given functions, ${w}_{n+11}=g\left(x,{x}_{1},...,{x}_{n}\right)$, is solved on $ℝ.$

## How to cite

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Tryhuk, Václav. "On transformations $z(t)=y(\varphi (t))$ of ordinary differential equations." Czechoslovak Mathematical Journal 50.3 (2000): 509-518. <http://eudml.org/doc/30579>.

@article{Tryhuk2000,
abstract = {The paper describes the general form of an ordinary differential equation of the order $n+1$$(n\ge 1) which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form $f\biggl (s, v, w\_\{11\}v\_\{1\}, \ldots , \sum \_\{j=1\}^\{n\}w\_\{nj\}v\_\{j\}\biggr ) = \sum \_\{j=1\}^\{n\}w\_\{n+1 j\}v\_\{j\} + w\_\{n+1 n+1\}f(x, v, v\_\{1\}, \ldots , v\_\{n\}),$ where w_\{ij\} = a_\{ij\}(x_\{1\}, \ldots , x_\{i-j+1\}) are given functions, w_\{n+1 1\} = g(x, x_\{1\}, \ldots , x_\{n\}), is solved on \mathbb \{R\}.}, author = {Tryhuk, Václav}, journal = {Czechoslovak Mathematical Journal}, keywords = {ordinary differential equations; linear differential equations; transformations; functional equations; ordinary differential equations; linear differential equations; transformations; functional equations}, language = {eng}, number = {3}, pages = {509-518}, publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic}, title = {On transformations z(t)=y(\varphi (t)) of ordinary differential equations}, url = {http://eudml.org/doc/30579}, volume = {50}, year = {2000}, } TY - JOUR AU - Tryhuk, Václav TI - On transformations z(t)=y(\varphi (t)) of ordinary differential equations JO - Czechoslovak Mathematical Journal PY - 2000 PB - Institute of Mathematics, Academy of Sciences of the Czech Republic VL - 50 IS - 3 SP - 509 EP - 518 AB - The paper describes the general form of an ordinary differential equation of the order n+1$$(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 $\mathbb {R}.$
LA - eng
KW - ordinary differential equations; linear differential equations; transformations; functional equations; ordinary differential equations; linear differential equations; transformations; functional equations
UR - http://eudml.org/doc/30579
ER -

## References

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7. 10.1023/A:1022414717364, Czechoslovak Math. J. 50 (125) (2000), 265–278. (2000) MR1761386DOI10.1023/A:1022414717364

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