On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

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May 2015, 35(5): 1767-1800. doi: 10.3934/dcds.2015.35.1767

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

Primitivo B. Acosta-Humánez ^1, , J. Tomás Lázaro ^2, , Juan J. Morales-Ruiz ^3, and Chara Pantazi ^2,

1.	Department of Mathematics, Universidad del Atlántico and Intelectual.Co, Barranquilla, Colombia
2.	Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, Spain, Spain
3.	Department of Applied Mathematics, Technical University of Madrid, Madrid, Spain

Received September 2013 Revised September 2014 Published December 2014

Abstract
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We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.

Keywords: Poincaré problem, rational first integral, Riccati equation, Differential Galois theory, integrating factor, Liouvillian solution., Liénard equation, Darboux theory of integrability.

Mathematics Subject Classification: Primary: 12H05; Secondary: 32S6.

Citation: Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767

References:

[1]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics,, PhD. Thesis, (2009). Google Scholar
[2]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrödinger Equation by Means of Differential Galois Theory,, VDM Verlag Dr. Müller, (2010). Google Scholar
[3]	P. B. Acosta-Humánez, La teoría de Morales-Ramis y el algoritmo de Kovacic,, Lecturas Matemáticas, 27 (2006), 21. Google Scholar
[4]	P. B. Acosta-Humánez, Nonautonomous Hamiltonian systems and Morales-Ramis theory I. The case $\ddot x=f(x,t)$,, SIAM Journal on Applied Dynamical Systems, 8 (2009), 279. doi: 10.1137/080730329. Google Scholar
[5]	P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some Hamiltonians with rational potential,, Discrete Continuous Dynam. Systems - B, 10 (2008), 265. doi: 10.3934/dcdsb.2008.10.265. Google Scholar
[6]	P. B. Acosta-Humánez, J. J. Morales-Ruiz and J.-A. Weil, Galoisian approach to integrability of Schrödinger equation,, Report on Mathematical Physics, 67 (2011), 305. doi: 10.1016/S0034-4877(11)60019-0. Google Scholar
[7]	P. B. Acosta-Humánez and C. Pantazi, Darboux integrals for Schrödinger planar vector fields via Darboux transformations,, SIGMA Symmetry Integrability Geom. Methods Appl., 8 (2012). doi: 10.3842/SIGMA.2012.043. Google Scholar
[8]	P. B. Acosta-Humánez, A. Reyes-Linero and J. Rodríguez-Contreras, Algebraic and qualitative remarks about the family $yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}$,, preprint 2014., (2014). Google Scholar
[9]	F. Baldassarri, On algebraic solutions of Lamé's differential equation,, J. Diff. Equat., 41 (1981), 44. doi: 10.1016/0022-0396(81)90052-8. Google Scholar
[10]	D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems,, VDM Verlag Dr. Müller, (2008). Google Scholar
[11]	D. Blázquez-Sanz and J.-J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems,, in Differential algebra, 509* (2010), 1. doi: 10.1090/conm/509. Google Scholar*
[12]	D. Blázquez-Sanz and Ch. Pantazi, A note on the Darboux theory of integrability of non-autonomous polynomial differential systems,, Nonlinearity, 25 (2012), 2615. doi: 10.1088/0951-7715/25/9/2615. Google Scholar
[13]	M. Carnicer, The Poincaré problem in the nondicritical case,, Ann. Math., 140 (1994), 289. doi: 10.2307/2118601. Google Scholar
[14]	G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres,, Ann. Inst. Fourier, 56 (2006), 735. doi: 10.5802/aif.2198. Google Scholar
[15]	D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbbC \mathbbP (2)$ having an invariant algebraic curve,, Ann. Inst. Fourier, 41 (1991), 883. doi: 10.5802/aif.1278. Google Scholar
[16]	E. S. Cheb-Terrab, Solutions for the general, confluent and biconfluent Heun equations and their connection with Abel equations,, J. Phys. A, 37 (2004), 9923. doi: 10.1088/0305-4470/37/42/007. Google Scholar
[17]	E. S. Cheb-Terrab and A. D. Roche, An Abel ordinary differential equation class generalizing known integrable classes,, European J. Appl. Math., 14 (2003), 217. doi: 10.1017/S0956792503005114. Google Scholar
[18]	T. S. Chihara, An Introduction to Orthogonal Polynomials,, Gordon and Breach Science Publishers, (1978). Google Scholar
[19]	C. Christopher, Invariant algebraic curves and conditions for a center,, Proc. Roy.Soc. Edinburgh Sect. A, 124 (1994), 1209. doi: 10.1017/S0308210500030213. Google Scholar
[20]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Darboux integrating factors: Inverse problems,, J. Diff. Equat., 250 (2011), 1. doi: 10.1016/j.jde.2010.10.013. Google Scholar
[21]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves,, Proc. Roy.Soc. Edinburgh Sect. A, 137 (2007), 1197. doi: 10.1017/S0308210506000400. Google Scholar
[22]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for invariant algebraic curves: Explicit computations,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 287. doi: 10.1017/S0308210507001175. Google Scholar
[23]	C. Christopher, J. Llibre, Ch. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems,, J. Phys. A, 35 (2002), 2457. doi: 10.1088/0305-4470/35/10/310. Google Scholar
[24]	C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields,, Pacific J. Math., 229 (2007), 63. doi: 10.2140/pjm.2007.229.63. Google Scholar
[25]	G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges),, Bull. Sci. math. 2ème série, 2 (1878), 60. Google Scholar
[26]	V. A. Dobrovol'skii, N. V. Lokot and J.-M. Strelcyn, Mikhail Nikolaevich Lagutinkskii (1871-1915): un mathématicien méconnu,, Historia Math., 25 (1998), 245. doi: 10.1006/hmat.1998.2194. Google Scholar
[27]	A. Duval and M. Loday-Richaud, Kovacic's algorithm and its application to some families of special functions,, Appl. Algebra Engrg. Comm. Comput., 3 (1992), 211. doi: 10.1007/BF01268661. Google Scholar
[28]	G. H. Halphen, Traité des Fonctions Elliptiques, Vol. I, II,, Gauthier-Villars, (1888). Google Scholar
[29]	I. A. García, H. Giacomini and J. Giné, Generalized nonlinear superposition principles for polynomial planar vector fields,, J. Lie Theory, 15 (2005), 89. Google Scholar
[30]	H. Giacomini , J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations,, Rocky Mountain J. Math., 36 (2006), 457. doi: 10.1216/rmjm/1181069462. Google Scholar
[31]	E. L. Ince, Ordinary Differential Equations,, Dover Publications, (1944). Google Scholar
[32]	J. P. Jouanolou, Equations de Pfaff Algébriques,, Lectures Notes in Mathematics, 708 (1979). doi: 10.1007/BFb0063393. Google Scholar
[33]	I. Kaplansky, An Introduction to Differential Algebra,, Hermann, (1957). Google Scholar
[34]	T. Kimura, On Riemann's equations which are solvable by quadratures,, Funkcialaj Ekvacioj, 12 (1969), 269. Google Scholar
[35]	J. Kovacic, An algorithm for solving second order linear homogeneous differential equations,, J. Symb. Comput., 2 (1986), 3. doi: 10.1016/S0747-7171(86)80010-4. Google Scholar
[36]	E. Kolchin, Differential Algebra and Algebraic Groups,, Academic Press, 54 (1973). Google Scholar
[37]	A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two,, J. Differential Geometry, 26 (1987), 1. Google Scholar
[38]	J. Llibre and Ch. Pantazi, Polynomial differential systems having a given Darbouxian first integral,, Bull. Sci. Math, 128 (2004), 775. doi: 10.1016/j.bulsci.2004.04.001. Google Scholar
[39]	J. Llibre and Ch. Pantazi, Darboux theory of integrability for a class of nonautonomous vector fields,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3205450. Google Scholar
[40]	J. Llibre, Ch. Pantazi and S. Walcher, Morphisms and inverse problems for Darboux integrating factors,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1291. doi: 10.1017/S0308210511001430. Google Scholar
[41]	J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems,, Ergod. Th. & Dynam. Sys., 31 (2011), 245. doi: 10.1017/S0143385709000868. Google Scholar
[42]	J. Llibre and X. Zhang, Darboux theory of integrability in $\C^n$ taking into account the multiplicity,, J. Diff. Equat., 246 (2009), 541. doi: 10.1016/j.jde.2008.07.020. Google Scholar
[43]	J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in $\mathbb R^n$ taking into account the multiplicity at infinity,, Bull. Sci. Math., 133 (2009), 765. doi: 10.1016/j.bulsci.2009.06.002. Google Scholar
[44]	B. Malgrange, Le groupoide de Galois d'un feuilletage,, in Essays on geometry and related topics, 38 (2001), 465. Google Scholar
[45]	B. Malgrange, On nonlinear differential Galois theory,, Chinese Ann. Math. Ser. B, 23 (2002), 219. doi: 10.1142/S0252959902000213. Google Scholar
[46]	J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resummation,, in Computer Algebra and Differential Equations, (1990), 117. Google Scholar
[47]	J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems,, Progress in Mathematics, 179 (1999). doi: 10.1007/978-3-0348-8718-2. Google Scholar
[48]	Ch. Pantazi, Inverse Problems of the Darboux Theory of Integrability for Planar Polynomial Differential Systems,, PhD. thesis, (2004). Google Scholar
[49]	J. V. Pereira, Vector fields, invariant varieties and linear systems,, Annales de l'institut Fourier, 51 (2001), 1385. doi: 10.5802/aif.1858. Google Scholar
[50]	J. V. Pereira, On the Poincaré problem for foliations of general type,, Math. Ann. , 323 (2002), 217. doi: 10.1007/s002080100277. Google Scholar
[51]	A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition,, Chapman and Hall, (2003). Google Scholar
[52]	E. G. C. Poole, Introduction to the Theory of Linear Differential Equations,, Dover Publications, (1960). Google Scholar
[53]	M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations,, Trans. Amer. Math. Soc., 279 (1983), 215. doi: 10.1090/S0002-9947-1983-0704611-X. Google Scholar
[54]	M. van der Put and M. Singer, Galois Theory of Linear Differential Equations,, Grundlehren der Mathematischen Wissenschaften, 328 (2003). doi: 10.1007/978-3-642-55750-7. Google Scholar
[55]	M. F. Singer, Liouvillian first integrals of differential equations,, Trans. Amer. Math. Soc., 333 (1992), 673. doi: 10.1090/S0002-9947-1992-1062869-X. Google Scholar
[56]	M. F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations,, Amer. J. Math., 103 (1981), 661. doi: 10.2307/2374045. Google Scholar
[57]	F. Ulmer and J. A. Weil., Note on Kovacic's algorithm,, J. Symb. Comp., 22 (1996), 179. doi: 10.1006/jsco.1996.0047. Google Scholar
[58]	R. Vidunas, Differential equations of order two with one singular point,, J. Symb. Comp., 28 (1999), 495. doi: 10.1006/jsco.1999.0312. Google Scholar
[59]	J.-A. Weil, Constantes et Polynômes de Darboux en Algèbre Differentielle: Applications Aux Systèmes Différentiels Linéaires,, PhD. Thesis, (1995). Google Scholar
[60]	J. A. Weil, Recent algorithms for solving second-order differential equations,, in Algorithms Seminar 2001-2002, (2003), 2001. Google Scholar
[61]	E. T. Whittaker and E. T. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions,, Cambridge Univ. Press, (1962). Google Scholar
[62]	H. Zołądek, Polynomial Riccati equations with algebraic solutions,, in Differential Galois theory (Bedlewo, 58 (2002), 219. doi: 10.4064/bc58-0-17. Google Scholar

show all references

References:

[1]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics,, PhD. Thesis, (2009). Google Scholar
[2]	P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrödinger Equation by Means of Differential Galois Theory,, VDM Verlag Dr. Müller, (2010). Google Scholar
[3]	P. B. Acosta-Humánez, La teoría de Morales-Ramis y el algoritmo de Kovacic,, Lecturas Matemáticas, 27 (2006), 21. Google Scholar
[4]	P. B. Acosta-Humánez, Nonautonomous Hamiltonian systems and Morales-Ramis theory I. The case $\ddot x=f(x,t)$,, SIAM Journal on Applied Dynamical Systems, 8 (2009), 279. doi: 10.1137/080730329. Google Scholar
[5]	P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some Hamiltonians with rational potential,, Discrete Continuous Dynam. Systems - B, 10 (2008), 265. doi: 10.3934/dcdsb.2008.10.265. Google Scholar
[6]	P. B. Acosta-Humánez, J. J. Morales-Ruiz and J.-A. Weil, Galoisian approach to integrability of Schrödinger equation,, Report on Mathematical Physics, 67 (2011), 305. doi: 10.1016/S0034-4877(11)60019-0. Google Scholar
[7]	P. B. Acosta-Humánez and C. Pantazi, Darboux integrals for Schrödinger planar vector fields via Darboux transformations,, SIGMA Symmetry Integrability Geom. Methods Appl., 8 (2012). doi: 10.3842/SIGMA.2012.043. Google Scholar
[8]	P. B. Acosta-Humánez, A. Reyes-Linero and J. Rodríguez-Contreras, Algebraic and qualitative remarks about the family $yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}$,, preprint 2014., (2014). Google Scholar
[9]	F. Baldassarri, On algebraic solutions of Lamé's differential equation,, J. Diff. Equat., 41 (1981), 44. doi: 10.1016/0022-0396(81)90052-8. Google Scholar
[10]	D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems,, VDM Verlag Dr. Müller, (2008). Google Scholar
[11]	D. Blázquez-Sanz and J.-J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems,, in Differential algebra, 509* (2010), 1. doi: 10.1090/conm/509. Google Scholar*
[12]	D. Blázquez-Sanz and Ch. Pantazi, A note on the Darboux theory of integrability of non-autonomous polynomial differential systems,, Nonlinearity, 25 (2012), 2615. doi: 10.1088/0951-7715/25/9/2615. Google Scholar
[13]	M. Carnicer, The Poincaré problem in the nondicritical case,, Ann. Math., 140 (1994), 289. doi: 10.2307/2118601. Google Scholar
[14]	G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres,, Ann. Inst. Fourier, 56 (2006), 735. doi: 10.5802/aif.2198. Google Scholar
[15]	D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbbC \mathbbP (2)$ having an invariant algebraic curve,, Ann. Inst. Fourier, 41 (1991), 883. doi: 10.5802/aif.1278. Google Scholar
[16]	E. S. Cheb-Terrab, Solutions for the general, confluent and biconfluent Heun equations and their connection with Abel equations,, J. Phys. A, 37 (2004), 9923. doi: 10.1088/0305-4470/37/42/007. Google Scholar
[17]	E. S. Cheb-Terrab and A. D. Roche, An Abel ordinary differential equation class generalizing known integrable classes,, European J. Appl. Math., 14 (2003), 217. doi: 10.1017/S0956792503005114. Google Scholar
[18]	T. S. Chihara, An Introduction to Orthogonal Polynomials,, Gordon and Breach Science Publishers, (1978). Google Scholar
[19]	C. Christopher, Invariant algebraic curves and conditions for a center,, Proc. Roy.Soc. Edinburgh Sect. A, 124 (1994), 1209. doi: 10.1017/S0308210500030213. Google Scholar
[20]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Darboux integrating factors: Inverse problems,, J. Diff. Equat., 250 (2011), 1. doi: 10.1016/j.jde.2010.10.013. Google Scholar
[21]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves,, Proc. Roy.Soc. Edinburgh Sect. A, 137 (2007), 1197. doi: 10.1017/S0308210506000400. Google Scholar
[22]	C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for invariant algebraic curves: Explicit computations,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 287. doi: 10.1017/S0308210507001175. Google Scholar
[23]	C. Christopher, J. Llibre, Ch. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems,, J. Phys. A, 35 (2002), 2457. doi: 10.1088/0305-4470/35/10/310. Google Scholar
[24]	C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields,, Pacific J. Math., 229 (2007), 63. doi: 10.2140/pjm.2007.229.63. Google Scholar
[25]	G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges),, Bull. Sci. math. 2ème série, 2 (1878), 60. Google Scholar
[26]	V. A. Dobrovol'skii, N. V. Lokot and J.-M. Strelcyn, Mikhail Nikolaevich Lagutinkskii (1871-1915): un mathématicien méconnu,, Historia Math., 25 (1998), 245. doi: 10.1006/hmat.1998.2194. Google Scholar
[27]	A. Duval and M. Loday-Richaud, Kovacic's algorithm and its application to some families of special functions,, Appl. Algebra Engrg. Comm. Comput., 3 (1992), 211. doi: 10.1007/BF01268661. Google Scholar
[28]	G. H. Halphen, Traité des Fonctions Elliptiques, Vol. I, II,, Gauthier-Villars, (1888). Google Scholar
[29]	I. A. García, H. Giacomini and J. Giné, Generalized nonlinear superposition principles for polynomial planar vector fields,, J. Lie Theory, 15 (2005), 89. Google Scholar
[30]	H. Giacomini , J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations,, Rocky Mountain J. Math., 36 (2006), 457. doi: 10.1216/rmjm/1181069462. Google Scholar
[31]	E. L. Ince, Ordinary Differential Equations,, Dover Publications, (1944). Google Scholar
[32]	J. P. Jouanolou, Equations de Pfaff Algébriques,, Lectures Notes in Mathematics, 708 (1979). doi: 10.1007/BFb0063393. Google Scholar
[33]	I. Kaplansky, An Introduction to Differential Algebra,, Hermann, (1957). Google Scholar
[34]	T. Kimura, On Riemann's equations which are solvable by quadratures,, Funkcialaj Ekvacioj, 12 (1969), 269. Google Scholar
[35]	J. Kovacic, An algorithm for solving second order linear homogeneous differential equations,, J. Symb. Comput., 2 (1986), 3. doi: 10.1016/S0747-7171(86)80010-4. Google Scholar
[36]	E. Kolchin, Differential Algebra and Algebraic Groups,, Academic Press, 54 (1973). Google Scholar
[37]	A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two,, J. Differential Geometry, 26 (1987), 1. Google Scholar
[38]	J. Llibre and Ch. Pantazi, Polynomial differential systems having a given Darbouxian first integral,, Bull. Sci. Math, 128 (2004), 775. doi: 10.1016/j.bulsci.2004.04.001. Google Scholar
[39]	J. Llibre and Ch. Pantazi, Darboux theory of integrability for a class of nonautonomous vector fields,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3205450. Google Scholar
[40]	J. Llibre, Ch. Pantazi and S. Walcher, Morphisms and inverse problems for Darboux integrating factors,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1291. doi: 10.1017/S0308210511001430. Google Scholar
[41]	J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems,, Ergod. Th. & Dynam. Sys., 31 (2011), 245. doi: 10.1017/S0143385709000868. Google Scholar
[42]	J. Llibre and X. Zhang, Darboux theory of integrability in $\C^n$ taking into account the multiplicity,, J. Diff. Equat., 246 (2009), 541. doi: 10.1016/j.jde.2008.07.020. Google Scholar
[43]	J. Llibre and X. Zhang, Darboux theory of integrability for polynomial vector fields in $\mathbb R^n$ taking into account the multiplicity at infinity,, Bull. Sci. Math., 133 (2009), 765. doi: 10.1016/j.bulsci.2009.06.002. Google Scholar
[44]	B. Malgrange, Le groupoide de Galois d'un feuilletage,, in Essays on geometry and related topics, 38 (2001), 465. Google Scholar
[45]	B. Malgrange, On nonlinear differential Galois theory,, Chinese Ann. Math. Ser. B, 23 (2002), 219. doi: 10.1142/S0252959902000213. Google Scholar
[46]	J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resummation,, in Computer Algebra and Differential Equations, (1990), 117. Google Scholar
[47]	J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems,, Progress in Mathematics, 179 (1999). doi: 10.1007/978-3-0348-8718-2. Google Scholar
[48]	Ch. Pantazi, Inverse Problems of the Darboux Theory of Integrability for Planar Polynomial Differential Systems,, PhD. thesis, (2004). Google Scholar
[49]	J. V. Pereira, Vector fields, invariant varieties and linear systems,, Annales de l'institut Fourier, 51 (2001), 1385. doi: 10.5802/aif.1858. Google Scholar
[50]	J. V. Pereira, On the Poincaré problem for foliations of general type,, Math. Ann. , 323 (2002), 217. doi: 10.1007/s002080100277. Google Scholar
[51]	A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition,, Chapman and Hall, (2003). Google Scholar
[52]	E. G. C. Poole, Introduction to the Theory of Linear Differential Equations,, Dover Publications, (1960). Google Scholar
[53]	M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations,, Trans. Amer. Math. Soc., 279 (1983), 215. doi: 10.1090/S0002-9947-1983-0704611-X. Google Scholar
[54]	M. van der Put and M. Singer, Galois Theory of Linear Differential Equations,, Grundlehren der Mathematischen Wissenschaften, 328 (2003). doi: 10.1007/978-3-642-55750-7. Google Scholar
[55]	M. F. Singer, Liouvillian first integrals of differential equations,, Trans. Amer. Math. Soc., 333 (1992), 673. doi: 10.1090/S0002-9947-1992-1062869-X. Google Scholar
[56]	M. F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations,, Amer. J. Math., 103 (1981), 661. doi: 10.2307/2374045. Google Scholar
[57]	F. Ulmer and J. A. Weil., Note on Kovacic's algorithm,, J. Symb. Comp., 22 (1996), 179. doi: 10.1006/jsco.1996.0047. Google Scholar
[58]	R. Vidunas, Differential equations of order two with one singular point,, J. Symb. Comp., 28 (1999), 495. doi: 10.1006/jsco.1999.0312. Google Scholar
[59]	J.-A. Weil, Constantes et Polynômes de Darboux en Algèbre Differentielle: Applications Aux Systèmes Différentiels Linéaires,, PhD. Thesis, (1995). Google Scholar
[60]	J. A. Weil, Recent algorithms for solving second-order differential equations,, in Algorithms Seminar 2001-2002, (2003), 2001. Google Scholar
[61]	E. T. Whittaker and E. T. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions,, Cambridge Univ. Press, (1962). Google Scholar
[62]	H. Zołądek, Polynomial Riccati equations with algebraic solutions,, in Differential Galois theory (Bedlewo, 58 (2002), 219. doi: 10.4064/bc58-0-17. Google Scholar

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