Geodesic
The moving frames for differential equations. II. Underdetermined and functional equations
Archivum mathematicum, Tome 40 (2004) no. 1, pp. 69-88
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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.
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.
Classification : 34A25, 34A26, 34C14, 34C41, 34K05, 34K17, 53B21
Keywords: pseudogroup; moving frame; equivalence of differential equations; differential equations with delay 
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Tryhuk, Václav; Dlouhý, Oldřich. The moving frames for differential equations. II. Underdetermined and functional equations. Archivum mathematicum, Tome 40 (2004) no. 1, pp. 69-88. http://geodesic.mathdoc.fr/item/ARM_2004_40_1_a7/

[1] Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York 1966. | MR

[2] Awane A., Goze M.: Pfaffian Systems, k–symplectic Systems. Kluwer Academic Publishers (Dordrecht–Boston–London), 2000. | MR | Zbl

[3] Bryant R., Chern S. S., Goldschmidt H., Griffiths P. A.: Exterior differential systems. Math. Sci. Res. Inst. Publ. 18, Springer - Verlag 1991. | MR | Zbl

[4] Cartan E.: Les systémes différentiels extérieurs et leurs applications géometriques. Hermann & Cie., Paris (1945). | MR | Zbl

[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).

[6] Čermák J.: Continuous transformations of differential equations with delays. Georgian Math. J. 2 (1995), 1–8. | MR | Zbl

[7] Chrastina J.: Transformations of differential equations. Equadiff 9 CD ROM, Papers, Masaryk University, Brno 1997, 83–92.

[8] Chrastina J.: The formal theory of differential equations. Folia Fac. Sci. Natur. Univ. Masaryk. Brun., Mathematica 6, 1998. | MR | Zbl

[9] Gardner R. B.: The method of equivalence and its applications. CBMS–NSF Regional Conf. Ser. in Appl. Math. 58, 1989. | MR | Zbl

[10] Neuman F.: On transformations of differential equations and systems with deviating argument. Czechoslovak Math. J. 31 (106) (1981), 87–90. | MR | Zbl

[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

[12] Neuman F.: Transformations and canonical forms of functional–differential equations. Proc. Roy. Soc. Edinburgh 115 A (1990), 349–357. | MR

[13] Neuman F.: Global Properties of Linear Ordinary Differential Equations. Math. Appl. (East European Series) 52, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991. | MR | Zbl

[14] Neuman F.: On equivalence of linear functional–differential equations. Results Math. 26 (1994), 354–359. | MR | Zbl

[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

[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

[17] Tryhuk V.: On global transformations of functional-differential equations of the first order. Czechoslovak Math. J. 50 (125) (2000), 279–293. | MR | Zbl

[18] Tryhuk V., Dlouhý O.: The moving frames for differential equations. I. The change of independent variable. Arch. Math. (Brno) 39 (2003), 317–333. | MR | Zbl