Geodesic
On global transformations of ordinary differential equations of the second order
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 499-508 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 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 $y\ne 0$, $v\ne 0.$
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 $y\ne 0$, $v\ne 0.$
Classification : 34-02, 34A25, 34A30, 34A34, 34C20, 39-02, 39B22, 39B40
Keywords: ordinary differential equations; linear differential equations; global transformations; functional equations 
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Tryhuk, Václav. On global transformations of ordinary differential equations of the second order. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 499-508. http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a3/

[n1] J. Aczél: Über Zusammenhänge zwischen Differential- und Funktionalgleichungen. Jahresber. Deutsch. Math.-Verein. 71 (1969), 55–57. | MR

[n3] J. Aczél: Lectures on Functional Equations and Their Applications. Academic Press, New York, 1966. | MR

[n4] M. Čadek: Form of general pointwise transformations of linear differential equations. Czechoslovak Math. J. 35 (110) (1985), 617–624. | MR

[n5] M. Čadek: Pointwise transformations of linear differential equations. Arch. Math. (Brno) 26 (1990), 187–200. | MR

[n2] A. Moór, L. Pintér: Untersuchungen Über den Zusammenhang von Differential- und Funktionalgleichungen. Publ. Math. Debrecen 13 (1966), 207–223. | MR

[n6] F. Neuman: Global Properties of Linear Ordinary Differential Equations. Mathematics and Its Applications (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. | MR | Zbl

[n7] F. Neuman: A note on smoothness of the Stäckel transformation. Prace Math. WSP Krakow 11 (1985), 147–151.

[n8] P. Stäckel: Über Transformationen von Differentialgleichungen. J. Reine Angew. Math. (Crelle Journal) 111 (1893), 290–302.

[n9] E. J. Wilczynski: Projective differential geometry of curves and ruled spaces. Teubner, Leipzig, 1906.