Abstract
The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) 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
where \(w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)\) for the given functions a ij is solved on \(\mathbb{R},u \ne {\text{0}}\).
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References
J. Aczél: Lectures on Functional Equations and Their Applications. Academic Press, New York, 1966.
J. Aczél: Über Zusammenhänge zwischen Differential-und Funktionalgleichungen. Jahresber. Deutsch. Math.-Verein. 71 (1969), 55–57.
O. Borůvka: Linear Differential Transformations of the Second Order. The English Univ. Press, London, 1971.
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.
J. Posluszny, L. A. Rubel: The motion of an ordinary differential equation. J. Differential Equations 34 (1979), 291–302.
V. Tryhuk: On transformations z(t) = y(ϕ(t)) of ordinary differential equations. Czechoslovak Math. J. 50(125) (2000), 509–518.
V. Tryhuk: On global transformations of ordinary differential equations of the second order. Czechoslovak Math. J. 50(125) (2000), 499–508.
V. Tryhuk: Remark to transformations of linear differential and functional-differential equations. Czechoslovak Math. J. 50(125) (2000), 265–278.
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Tryhuk, V. Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations. Czechoslovak Mathematical Journal 50, 519–529 (2000). https://doi.org/10.1023/A:1022829525930
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DOI: https://doi.org/10.1023/A:1022829525930