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

Czechoslovak Mathematical Journal (2000)

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

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topTryhuk, 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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- Linear Differential Transformations of the Second Order, The English Univ. Press, London, 1971. (1971) MR0463539
- Untersuchungen Über den Zusammenhang von Differential- und Funktionalgleichungen, Publ. Math. Debrecen 13 (1966), 207–223. (1966) MR0206445
- Global Properties of Linear Ordinary Differential Equations, Mathematics and Its Applications (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. (1991) Zbl0784.34009MR1192133
- 10.1016/0022-0396(79)90011-1, J. Differential Equations 34 (1979), 291–302. (1979) MR0550047DOI10.1016/0022-0396(79)90011-1
- 10.1023/A:1022414717364, Czechoslovak Math. J. 50 (125) (2000), 265–278. (2000) MR1761386DOI10.1023/A:1022414717364

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