Geodesic
On Differential Invariants and Classification of Ordinary Differential Equations of the Form $y''=A(x,y)y'+B(x,y)$
Matematičeskie zametki, Tome 104 (2018) no. 2, pp. 163-173.

Voir la notice de l'article provenant de la source Math-Net.Ru

The class of second-order ordinary differential equations $y''=A(x,y)y'+B(x,y)$ is studied by methods of the geometry of jet spaces and the geometric theory of differential equations. The symmetry group of this class of equations is calculated, and the field of differential invariants of its action on equations is described. These results are used to state and prove a criterion for the local equivalence of two nondegenerate ordinary differential equations of the form $y''=A(x,y)y'+B(x,y)$, in which the coefficients $A$ and $B$ are rational in $x$ and $y$.
Keywords: ordinary differential equation, symmetry group, differential invariant.
Mots-clés : jet space 
@article{MZM_2018_104_2_a0,
     author = {P. V. Bibikov},
     title = {On {Differential} {Invariants} and {Classification} of {Ordinary} {Differential} {Equations} of the {Form} $y''=A(x,y)y'+B(x,y)$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {163--173},
     publisher = {mathdoc},
     volume = {104},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a0/}
}
TY  - JOUR
AU  - P. V. Bibikov
TI  - On Differential Invariants and Classification of Ordinary Differential Equations of the Form $y''=A(x,y)y'+B(x,y)$
JO  - Matematičeskie zametki
PY  - 2018
SP  - 163
EP  - 173
VL  - 104
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a0/
LA  - ru
ID  - MZM_2018_104_2_a0
ER  - 
%0 Journal Article
%A P. V. Bibikov
%T On Differential Invariants and Classification of Ordinary Differential Equations of the Form $y''=A(x,y)y'+B(x,y)$
%J Matematičeskie zametki
%D 2018
%P 163-173
%V 104
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a0/
%G ru
%F MZM_2018_104_2_a0
P. V. Bibikov. On Differential Invariants and Classification of Ordinary Differential Equations of the Form $y''=A(x,y)y'+B(x,y)$. Matematičeskie zametki, Tome 104 (2018) no. 2, pp. 163-173. http://geodesic.mathdoc.fr/item/MZM_2018_104_2_a0/

[1] B. S. Kruglikov, “Point classification of second order ODEs: Tresse classification revisited and beyond”, Differential Equations: Geometry, Symmetries and Integrability, Springer, Berlin, 2009 | MR | Zbl

[2] A. Tresse, Détermination des invariants ponctuels de l'équation différentielle ordinaire du second ordre $y''=\omega(x,y,y')$, Leipzig, 1896 | Zbl

[3] R. A. Sharipov, Effective Procedure of Point Classification for the Equation $y''=P+3Qy'+3Ry'^2+Sy'^3$, 1998, arXiv: 9802027

[4] S. Lie, “Klassification und Integration von gewöhnlichen Differentialgleichungen zwischen $x$, $y$, die eine Gruppe von Transformationen gestatten. III”, Lie Arch., 8 (1883), 371–458 | Zbl

[5] R. Liouville, “Sur les invariants de certaines eq́uations diffeŕentielles et sur leurs applications”, J. de l'Ećole Polytech., 59 (1889), 7–76 | Zbl

[6] S. Yu. Slavyanov, “O ponizhenii polinomialnoi stepeni fuksovoi ($2\times 2$)-sistemy”, TMF, 182:2 (2015), 223–230 | DOI | MR | Zbl

[7] B. Dubrov, “Contact trivialization of ordinary differential equations”, Differential Geometry and Its Applications, Silesian Univ. Opava, Opava, 2001, 73–84 | MR | Zbl

[8] B. Dubrov, “Ob odnom klasse kontaktnykh invariantov sistem obyknovennykh differentsialnykh uravnenii”, Izv. vuzov. Matem., 2006, no. 1, 76–77 | MR | Zbl

[9] A. Kushner, Klassifikatsi uravnenii Monzha–Ampera, Dis. $\dots$ dokt. fiz.-matem. nauk, Kazan, 2011

[10] A. Kushner, V. Lychagin, V. Rubtsov, Contact Geometry and Non-Linear Differential Equations, Cambridge Univ. Press, Cambridge, 2007 | MR

[11] D. V. Alekseevskii, A. M. Vinogradov, V. V. Lychagin, “Osnovnye idei i ponyatiya differentsialnoi geometrii”, Geometriya – 1, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 28, VINITI, M., 1988, 5–289 | MR | Zbl

[12] A. M. Vinogradov, I. S. Krasilschik, V. V. Lychagin, Vvedenie v geometriyu nelineinykh differentsialnykh uravnenii, Nauka, M., 1986 | MR

[13] P. V. Bibikov, “Zadacha Li i differentsialnye invarianty ODU vida $y''=F(x,y)$”, Funkts. analiz i ego pril., 51:4 (2017), 16–25 | DOI

[14] P. Bibikov, A. Malakhov, “On Lie problem and differential invariants for the subgroup of the plane Cremona group”, J. Geom. Phys., 121 (2017), 72–82 | MR | Zbl

[15] P. Bibikov, “Generalized Lie problem and differential invariants for the third order ODEs”, Lobachevskii J. Math., 38:4 (2017), 622–629 | DOI | MR | Zbl

[16] V. I. Arnold, Geometricheskie metody v teorii obyknovennykh differentsialnykh uravnenii, RKhD, Izhevsk, 2000 | MR | Zbl

[17] E. Cartan, “Sur les variétés à connexion projective”, Bull. Soc. Math. France, 52 (1924), 205–241 | DOI | MR | Zbl

[18] V. A. Yumaguzhin, Differential Invariants of $2$-order ODEs, I, arXiv: 0804.0674

[19] B. Kriglikov, V. Lychagin, “Global Lie–Tresse theorem”, Selecta Math. (N.S.), 22:3 (2016), 1357–1411 | DOI | MR | Zbl

[20] P. V. Bibikov, “O tochechnoi ekvivalentnosti funktsii na prostranstve $1$-dzhetov $J^1\mathbb{R}$”, Funkts. analiz i ego pril., 48:4 (2014), 19–25 | DOI | MR | Zbl

[21] E. Vinberg, V. Popov, Teoriya invariantov, VINITI, M., 1989

[22] M. Rosenlicht, “A remark on quotient spaces”, An. Acad. Brasil. Ci., 35 (1963), 487–489 | MR | Zbl

[23] B. Bukhberger, Kompyuternaya algebra. Simvolnye i algebraicheskie vychisleniya, Mir, M., 1986 | Zbl

[24] V. V. Lychagin, “Feedback differential invariants”, Acta Appl. Math., 109:1 (2010), 211–222 | DOI | MR | Zbl