×

Equivalence of third-order ordinary differential equations to Chazy equations I-XIII. (English) Zbl 1196.34047

Summary: We solve the equivalence problem of a third-order ordinary differential equations quadratic in the second-order derivative. For this class of equations the invariants of the group of point equivalence transformations and the invariant differentiation operators are constructed. Using these results the invariants of 13 Chazy equations were calculated. We provide examples of finding equivalent equations by use of their invariants. Also two new examples of the equations linearizable by a local transformation are found. These are a particular case of Chazy – XII equation and a Schwarzian equation.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C14 Symmetries, invariants of ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1111/1467-9590.00130 · Zbl 1136.34350 · doi:10.1111/1467-9590.00130
[2] DOI: 10.1137/1018074 · Zbl 0329.35015 · doi:10.1137/1018074
[3] Olver P. J., Lecture Notes in Physics 195 pp 273– (1984)
[4] DOI: 10.1063/1.524491 · Zbl 0445.35056 · doi:10.1063/1.524491
[5] Ince E. L., Ordinary Differential Equations (1956) · Zbl 0063.02971
[6] DOI: 10.1007/BF02393131 · JFM 42.0340.03 · doi:10.1007/BF02393131
[7] DOI: 10.1007/BF02412437 · Zbl 0129.06101 · doi:10.1007/BF02412437
[8] DOI: 10.1111/1467-9590.00134 · Zbl 1136.34351 · doi:10.1111/1467-9590.00134
[9] DOI: 10.1111/j.1467-9590.2006.00346.x · Zbl 1145.34379 · doi:10.1111/j.1467-9590.2006.00346.x
[10] DOI: 10.1088/0305-4470/32/45/309 · Zbl 0943.34085 · doi:10.1088/0305-4470/32/45/309
[11] DOI: 10.2991/jnmp.2002.9.3.4 · Zbl 1028.34082 · doi:10.2991/jnmp.2002.9.3.4
[12] Ibragimov N. H., Notices South Afric. Math. Soc. 29 pp 61– (1997)
[13] Cartan E., Oeuvres Completes 2 (1953)
[14] Gardner R. B., The Method of Equivalence and its Applications (1989) · Zbl 0694.53027 · doi:10.1137/1.9781611970135
[15] Olver P. J., Equivalence, invariants and symmetry, in Graduate Texts in Mathematics (1995) · Zbl 0837.58001 · doi:10.1017/CBO9780511609565
[16] Ibragimov N. H., Arch. ALGA 1 pp 9– (2004)
[17] 17. R. A.Sharipov, Effective procedure of point-classification for the equationsy” =P+ 3Qy’ + 3Ry’2+Sy’3, Electronic archive at LANL math.DG/9802027, 1998 .
[18] Kamran N., J. Diff. Geometry 22 pp 139– (1985)
[19] DOI: 10.1134/S0012266107050035 · Zbl 1170.34320 · doi:10.1134/S0012266107050035
[20] Lie S., Geometrie der Berurungstransformationen (1896)
[21] Ovsiannikov L. V., Group Analysis of Differential Equations (1982) · Zbl 0485.58002
[22] Neut S., C. R. Acad. Sci. Paris, Ser. I 335 pp 515– (2002) · Zbl 1016.34007 · doi:10.1016/S1631-073X(02)02507-4
[23] DOI: 10.1016/j.jmaa.2005.01.025 · Zbl 1082.34003 · doi:10.1016/j.jmaa.2005.01.025
[24] DOI: 10.1063/1.528613 · Zbl 0698.35137 · doi:10.1063/1.528613
[25] DOI: 10.1063/1.527129 · Zbl 0598.35117 · doi:10.1063/1.527129
[26] DOI: 10.1002/cpa.3160300106 · Zbl 0338.35024 · doi:10.1002/cpa.3160300106
[27] DOI: 10.1063/1.525875 · Zbl 0531.35069 · doi:10.1063/1.525875
[28] Polyanin A. D., Handbook of Exact Solutions for Ordinary Differential Equations (1995) · Zbl 0877.34001
[29] Hille E., Ordinary Differential Equations in the Complex Domain (1976) · Zbl 0343.34007
[30] Musette M., Painleve Property: One Century Later (1999)
[31] Hirota R., J. Phys. Soc. Japan 40 pp 611– (1976)
[32] Ablowitz M. J., Stud. Appl. Math. 53 pp 249– (1974) · Zbl 0408.35068 · doi:10.1002/sapm1974534249
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.