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
is solved on \(\mathbb{R}{\text{ for }}y \ne 0,\upsilon \ne 0\).
Similar content being viewed by others
References
J. Aczél: Über Zusammenhänge zwischen Differential-und Funktionalgleichungen. Jahresber. Deutsch. Math.-Verein. 71 (1969), 55–57.
J. Aczél: Lectures on Functional Equations and Their Applications. Academic Press, New York, 1966.
M. Čadek: Form of general pointwise transformations of linear differential equations. Czechoslovak Math. J. 35(110) (1985), 617–624.
M. Čadek: Pointwise transformations of linear differential equations. Arch. Math. (Brno) 26 (1990), 187–200.
A. Moór, L. Pintér: Untersuchungen Über den Zusammenhang von Differential-und Funktionalgleichungen. Publ. Math. Debrecen 13 (1966), 207–223.
F. Neuman: Global Properties of Linear Ordinary Differential Equations. Mathematics and Its Applications (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991.
F. Neuman: A note on smoothness of the Stäckel transformation. Prace Math. WSP Krakow 11 (1985), 147–151.
P. Stäckel: Über Transformationen von Differentialgleichungen. J. Reine Angew. Math. (Crelle Journal) 111 (1893), 290–302.
E. J. Wilczynski: Projective differential geometry of curves and ruled spaces. Teubner, Leipzig, 1906.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tryhuk, V. On global transformations of ordinary differential equations of the second order. Czechoslovak Mathematical Journal 50, 499–508 (2000). https://doi.org/10.1023/A:1022825325021
Issue Date:
DOI: https://doi.org/10.1023/A:1022825325021