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A reflecting function of a family of functions. (English. Russian original) Zbl 1001.34027
Differ. Equ. 36, No. 12, 1794-1800 (2000); translation from Differ. Uravn. 36, No. 12, 1636-1641 (2000).
The author, pursuing his previous work [Reflecting function and periodic solutions to differential equations. (Russian). Minsk: Izdatel’stvo “Universitetskoe” (1986; Zbl 0607.34038)], develops a method that permits to study solutions to differential equations related by a transformation involving a time reflection \(t \to - t\). Such mapping, which could be seen as a “reversing symmetry” [see, e.g., M. B. Sevryuk, Reversible systems, LNM 1211. Berlin etc.: Springer-Verlag (1986; Zbl 0661.58002)] is not necessarily applicable to all solutions, but could be restricted to a suitable subclass, as in the Levi-Winternitz theory of conditional symmetries [D. Levi and P. Winternitz, J. Phys. A, Math. Gen. 22, No. 15, 2915-2924 (1989; Zbl 0694.35159) and G. Cicogna and G. Gaeta, J. Phys. A, Math. Gen. 34, No. 3, 491-512 (2001; Zbl 0969.35010)]. Several special classes of equations and solutions are considered.

MSC:
34C14 Symmetries, invariants of ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
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