# Article

 Title: The moving frames for differential equations. II. Underdetermined and functional equations (English) Author: Tryhuk, Václav Author: Dlouhý, Oldřich Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 40 Issue: 1 Year: 2004 Pages: 69-88 Summary lang: English . Category: math . Summary: Continuing the idea of Part I, we deal with more involved pseudogroup of transformations $\bar{x}=\varphi (x)$, $\bar{y}=L(x)y$, $\bar{z}=M(x)z,\, \ldots$ applied to the first order differential equations including the underdetermined case (i.e. the Monge equation $y^{\prime }=f(x,y,z,z^{\prime })$) and certain differential equations with deviation (if $z=y(\xi (x))$ is substituted). Our aim is to determine complete families of invariants resolving the equivalence problem and to clarify the largest possible symmetries. Together with Part I, this article may be regarded as an introduction into the method of moving frames adapted to the theory of differential and functional-differential equations. (English) Keyword: pseudogroup Keyword: moving frame Keyword: equivalence of differential equations Keyword: differential equations with delay MSC: 34A25 MSC: 34A26 MSC: 34C14 MSC: 34C41 MSC: 34K05 MSC: 34K17 MSC: 53B21 idZBL: Zbl 1117.34058 idMR: MR2054874 . Date available: 2008-06-06T22:43:00Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107892 . Related article: http://dml.cz/handle/10338.dmlcz/107880 . Reference: [1] Aczél J.: Lectures on Functional Equations and Their Applications.Academic Press, New York 1966. MR 0208210 Reference: [2] Awane A., Goze M.: Pfaffian Systems, k–symplectic Systems.Kluwer Academic Publishers (Dordrecht–Boston–London), 2000. Zbl 0957.58004, MR 1779116 Reference: [3] Bryant R., Chern S. S., Goldschmidt H., Griffiths P. A.: Exterior differential systems.Math. Sci. Res. Inst. Publ. 18, Springer - Verlag 1991. Zbl 0726.58002, MR 1083148 Reference: [4] Cartan E.: Les systémes différentiels extérieurs et leurs applications géometriques.Hermann & Cie., Paris (1945). Zbl 0063.00734, MR 0016174 Reference: [5] Cartan E.: Sur la structure des groupes infinis de transformations.Ann. Ec. Norm. 3-e serie, t. XXI, 1904 (also Oeuvres Complètes, Partie II, Vol 2, Gauthier–Villars, Paris 1953). Reference: [6] Čermák J.: Continuous transformations of differential equations with delays.Georgian Math. J. 2 (1995), 1–8. Zbl 0817.34036, MR 1310496 Reference: [7] Chrastina J.: Transformations of differential equations.Equadiff 9 CD ROM, Papers, Masaryk University, Brno 1997, 83–92. Reference: [8] Chrastina J.: The formal theory of differential equations.Folia Fac. Sci. Natur. Univ. Masaryk. Brun., Mathematica 6, 1998. Zbl 0906.35002, MR 1656843 Reference: [9] Gardner R. B.: The method of equivalence and its applications.CBMS–NSF Regional Conf. Ser. in Appl. Math. 58, 1989. Zbl 0694.53027, MR 1062197 Reference: [10] Neuman F.: On transformations of differential equations and systems with deviating argument.Czechoslovak Math. J. 31 (106) (1981), 87–90. Zbl 0463.34051, MR 0604115 Reference: [11] Neuman F.: Simultaneous solutions of a system of Abel equations and differential equations with several delays.Czechoslovak Math. J. 32 (107) (1982), 488–494. MR 0669790 Reference: [12] Neuman F.: Transformations and canonical forms of functional–differential equations.Proc. Roy. Soc. Edinburgh 115 A (1990), 349–357. MR 1069527 Reference: [13] Neuman F.: Global Properties of Linear Ordinary Differential Equations.Math. Appl. (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. Zbl 0784.34009, MR 1192133 Reference: [14] Neuman F.: On equivalence of linear functional–differential equations.Results Math. 26 (1994), 354–359. Zbl 0829.34054, MR 1300618 Reference: [15] Tryhuk V.: The most general transformations of homogeneous linear differential retarded equations of the first order.Arch. Math. (Brno) 16 (1980), 225–230. MR 0594470 Reference: [16] Tryhuk V.: The most general transformation of homogeneous linear differential retarded equations of the $n$-th order.Math. Slovaca 33 (1983), 15–21. MR 0689272 Reference: [17] Tryhuk V.: On global transformations of functional-differential equations of the first order.Czechoslovak Math. J. 50 (125) (2000), 279–293. Zbl 1054.34105, MR 1761387 Reference: [18] Tryhuk V., Dlouhý O.: The moving frames for differential equations. I. The change of independent variable.Arch. Math. (Brno) 39 (2003), 317–333. Zbl 1116.34301, MR 2032105 .

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