Geodesic
On global transformations of ordinary differential equations of the second order
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 499-508.

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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 $y\ne 0$, $v\ne 0.$
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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     title = {On global transformations of ordinary differential equations of the second order},
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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/