# Article

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Keywords:
moving coframe; equivalence of differential equations; symmetry of differential equations; differential invariant; Maurer-Cartan form
Summary:
The article concerns the symmetries of differential equations with short digressions to the underdetermined case and the relevant differential equations with delay. It may be regarded as an introduction into the method of moving frames relieved of the geometrical aspects: the stress is made on the technique of calculations employing only the most fundamental properties of differential forms. The present Part I is devoted to a single ordinary differential equation subjected to the change of the independent variable, the unknown function is preserved.
References:
[1] Aczél J.: Über Zusammenhänge zwischen Differential– und Funktionalgleichungen. Jahresber. Deutsch. Math. Ver. 71 (1969), 55–57. MR 0256014 | Zbl 0175.45603
[2] Awane A., Goze M.: Pfaffian Systems, k-symplectic Systems. Kluwer Academic Publischers (Dordrecht–Boston–London), 2000. MR 1779116 | Zbl 0957.58004
[3] Borůvka O.: Linear Differential Transformations of the Second Order. The English Univ. Press, London, 1971. MR 0463539
[4] Bryant R., Chern S. S., Goldschmidt H., Griffiths P. A.: Exterior differential systems. Mat. Sci. Res. Inst. Publ. 18, Springer-Verlag 1991. MR 1083148 | Zbl 0726.58002
[5] Cartan E.: Les systémes différentiels extérieurs et leurs applications géometriques. Act. Scient. et Ind. 994 (1945). MR 0016174 | Zbl 0063.00734
[6] 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).
[7] Chrastina J.: Transformations of differential equations. Equadiff 9 CD ROM, Papers, Masaryk univerzity, Brno 1997, 83–92.
[8] Chrastina J.: The formal theory of differential equations. Folia Fac. Scient. Nat. Univ. Masarykianae Brunensis, Mathematica 6, 1998. MR 1656843 | Zbl 0906.35002
[9] Gardner R. B.: The method of equivalence and its applications. CBMS–NSF Regional Conf. in Appl. Math. 58, 1989. MR 1062197 | Zbl 0694.53027
[10] Moór A., Pintér L.: Untersuchungen über den Zusammenhang von Differential– und Funktionalgleichungen. Publ. Math. Debrecen 13 (1966), 207–223. MR 0206445 | Zbl 0199.15301
[11] Neuman F.: Global Properties of Linear Ordinary Differential Equations. Mathematics and Its Applications (East European Series) 52, Kluwer Acad. Publ., Dordrecht–Boston–London, 1991. MR 1192133 | Zbl 0784.34009
[12] Posluszny J., Rubel L. A.: The motion of an ordinary differential equation. J. Differential Equations 34 (1979), 291–302. MR 0550047
[13] Sharpe R. V.: Differential geometry. Graduate Texts in Math. 166, Springer Verlag, 1997. MR 1453120 | Zbl 0876.53001
[14] Tryhuk V.: On transformations $z(t)=y(\varphi (t))$ of ordinary differential equations. Czech. Math. J., 50(125) (2000), Praha, 509–518. MR 1777472 | Zbl 1079.34505

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