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A simple method for constructing non-liouvillian first integrals of autonomous planar systems
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 987-999 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.
We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.
Classification : 33C99, 34A25, 34C07, 34C14, 81U15
Keywords: planar polynomial systems; Kukles systems; generalized Liénard systems; non-liouvillian first integrals 
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Schulze-Halberg, Axel. A simple method for constructing non-liouvillian first integrals of autonomous planar systems. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 987-999. http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a17/

[1] M.  Bronstein and Sébastien Lafaille: Solutions of linear ordinary differential equations in terms of special functions. In: Proceedings of the 2002  International Symposium on Symbolic and Algebraic Computation, , , 2002, pp. 23–28. | MR

[2] L. A. Čerkas: Conditions for a center for a certain Liénard equation. Differencial’nye Uravnenija 12 (1976), 292–298, 379. | MR

[3] J.  Chavarriga, I. A.  García, and J. Giné: On Lie’s symmetries for planar polynomial differential systems. Nonlinearity 14 (2001), 863–880. | DOI | MR

[4] E. S.  Cheb-Terrab: Non-Liouvillian solutions for second order linear ODEs. Proceedings of the Workshop on Group Theory and Numerical Analysis, CRM, Montreal, 2003. | MR

[5] G.  Darboux: Mémoire sur les équations différentielles algébraiques du premier ordre et du premier degré (Mélanges). Bull. Sci. Math. 2 (1878), 60–96, 123–144, 151–200.

[6] H.  Giacomini, J.  Giné, and M.  Grau: Integrability of planar polynomial differential systems through linear differential equations. Rocky Mountain J.  Math. (2004), . | MR

[7] J.  Giné: Conditions for the existence of a center for the Kukles homogeneous system. Comput. Math. Appl. 43 (2002), 1261–1269. | DOI | MR

[8] I. S.  Gradshteyn, I. M.  Ryzhik: Tables of Series, Products and Integrals. Academic Press, San Diego, 1994. | MR

[9] A.  Goriely: Integrability and Nonintegrability of Dynamical Systems. World Scientific Publishing, Singapore, 2001. | MR | Zbl

[10] C.  Grosche, F.  Steiner: How to solve path integrals in quantum mechanics. J.  Math. Phys. 36 (1995), 2354–2385. | DOI | MR

[11] M.  Hayashi: On the local center of generalized Liénard-type systems. Far East J.  Math. Sci. (FJMS) 10 (2003), 265–283. | MR | Zbl

[12] D.  Hilbert: Mathematical problems. Reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479. | DOI | MR

[13] P. D.  Lax: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467–490. | DOI | MR | Zbl

[14] Z.  Liu, E.  Sáez, and I.  Szántó: A cubic system with an invariant triangle surrounding at least one limit cycle. Taiwanese J.  Math. 7 (2003), 275–281. | DOI | MR

[15] R.  Milson: Liouville transformation and exactly solvable Schrödinger equations. Internat. J.  Theor. Phys. 37 (1998), 1735–1752. | DOI | MR | Zbl

[16] M. J.  Prelle, M. F.  Singer: Elementary first integrals of differential equations. Trans. Am. Math. Soc. 279 (1983), 215–229. | DOI | MR

[17] M. F.  Singer: Liouvillian first integrals of differential equations. Trans. Amer. Math. Soc. 333 (1992), 673–688. | DOI | MR | Zbl

[18] J.-M.  Strelcyn, S.  Wojciechowski: A method of finding integrals for three-dimensional dynamical systems. Phys. Lett. A 133 (1988), 207–212. | DOI | MR

[19] A. G.  Ushveridze: Quasi-exactly Solvable Models in Quantum Mechanics. Institute of Physics Publishing, London, 1994. | MR | Zbl

[20] F. J. Wright: Recognising and solving special function ODEs. Preprint (2000).

[21] C.-D.  Zhao, Q.-M.  He: On the center of the generalized Liénard system. Czechoslovak Math.  J. 52 (127) (2002), 817–832. | DOI | MR | Zbl