On global transformations of ordinary differential equations of the second order

Václav Tryhuk

Czechoslovak Mathematical Journal (2000)

  • Volume: 50, Issue: 3, page 499-508
  • ISSN: 0011-4642

Abstract

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The paper describes the general form of an ordinary differential equation of the second order 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 ( t , v y , w y + u v z ) = f ( x , y , z ) u 2 v + g ( t , x , u , v , w ) v z + h ( t , x , u , v , w ) y + 2 u w z is solved on ℝ for y ≠ 0 , v ≠ 0 .

How to cite

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Tryhuk, Václav. "On global transformations of ordinary differential equations of the second order." Czechoslovak Mathematical Journal 50.3 (2000): 499-508. <http://eudml.org/doc/30578>.

@article{Tryhuk2000,
abstract = {The paper describes the general form of an ordinary differential equation of the second order 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 \[ f(t,vy,wy+uvz)=f(x,y,z)u^\{2\}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz \] is solved on $\mathbb \{R\}$ for $y\ne 0$, $v\ne 0.$},
author = {Tryhuk, Václav},
journal = {Czechoslovak Mathematical Journal},
keywords = {ordinary differential equations; linear differential equations; global transformations; functional equations; ordinary differential equations; linear differential equations; global transformations; functional equations},
language = {eng},
number = {3},
pages = {499-508},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On global transformations of ordinary differential equations of the second order},
url = {http://eudml.org/doc/30578},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Tryhuk, Václav
TI - On global transformations of ordinary differential equations of the second order
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 3
SP - 499
EP - 508
AB - The paper describes the general form of an ordinary differential equation of the second order 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 \[ f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz \] is solved on $\mathbb {R}$ for $y\ne 0$, $v\ne 0.$
LA - eng
KW - ordinary differential equations; linear differential equations; global transformations; functional equations; ordinary differential equations; linear differential equations; global transformations; functional equations
UR - http://eudml.org/doc/30578
ER -

References

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  2. Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966. (1966) MR0208210
  3. Form of general pointwise transformations of linear differential equations, Czechoslovak Math. J. 35 (110) (1985), 617–624. (1985) MR0809044
  4. Pointwise transformations of linear differential equations, Arch. Math. (Brno) 26 (1990), 187–200. (1990) MR1188970
  5. Untersuchungen Über den Zusammenhang von Differential- und Funktionalgleichungen, Publ. Math. Debrecen 13 (1966), 207–223. (1966) MR0206445
  6. 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
  7. A note on smoothness of the Stäckel transformation, Prace Math. WSP Krakow 11 (1985), 147–151. (1985) 
  8. Über Transformationen von Differentialgleichungen, J. Reine Angew. Math. (Crelle Journal) 111 (1893), 290–302. (1893) 
  9. Projective differential geometry of curves and ruled spaces, Teubner, Leipzig, 1906. (1906) 

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