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On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory
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 |
References:
[1] |
P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics, PhD. Thesis, Universitat Politecnica de Catalunya, 2009, arXiv:0906.3532. |
[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, Berlin, 2010. |
[3] |
P. B. Acosta-Humánez, La teoría de Morales-Ramis y el algoritmo de Kovacic, Lecturas Matemáticas, 27 (2006), 21-56. |
[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), 279-297.
doi: 10.1137/080730329. |
[5] |
P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some Hamiltonians with rational potential, Discrete Continuous Dynam. Systems - B, 10 (2008), 265-293.
doi: 10.3934/dcdsb.2008.10.265. |
[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), 305-374.
doi: 10.1016/S0034-4877(11)60019-0. |
[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), Paper 043, 26 pp.
doi: 10.3842/SIGMA.2012.043. |
[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. |
[9] |
F. Baldassarri, On algebraic solutions of Lamé's differential equation, J. Diff. Equat., 41 (1981), 44-58.
doi: 10.1016/0022-0396(81)90052-8. |
[10] |
D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems, VDM Verlag Dr. Müller, Berlin, 2008. |
[11] |
D. Blázquez-Sanz and J.-J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in Differential algebra, complex analysis and orthogonal polynomials, (eds P.B. Acosta-Humánez and F. Marcellán), Contemp. Math., Amer. Math. Soc., Providence, 509 (2010), 1-58.
doi: 10.1090/conm/509. |
[12] |
D. Blázquez-Sanz and Ch. Pantazi, A note on the Darboux theory of integrability of non-autonomous polynomial differential systems, Nonlinearity, 25 (2012), 2615-2624.
doi: 10.1088/0951-7715/25/9/2615. |
[13] |
M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math., 140 (1994), 289-294.
doi: 10.2307/2118601. |
[14] |
G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres, Ann. Inst. Fourier, 56 (2006), 735-779.
doi: 10.5802/aif.2198. |
[15] |
D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbbC \mathbbP (2)$ having an invariant algebraic curve, Ann. Inst. Fourier, 41 (1991), 883-903.
doi: 10.5802/aif.1278. |
[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), 9923-9949.
doi: 10.1088/0305-4470/37/42/007. |
[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), 217-229.
doi: 10.1017/S0956792503005114. |
[18] |
T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. |
[19] |
C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy.Soc. Edinburgh Sect. A, 124 (1994), 1209-1229.
doi: 10.1017/S0308210500030213. |
[20] |
C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Darboux integrating factors: Inverse problems, J. Diff. Equat., 250 (2011), 1-25.
doi: 10.1016/j.jde.2010.10.013. |
[21] |
C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves, Proc. Roy.Soc. Edinburgh Sect. A, 137 (2007), 1197-1226.
doi: 10.1017/S0308210506000400. |
[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), 287-302.
doi: 10.1017/S0308210507001175. |
[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), 2457-2476.
doi: 10.1088/0305-4470/35/10/310. |
[24] |
C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229 (2007), 63-117.
doi: 10.2140/pjm.2007.229.63. |
[25] |
G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200. |
[26] |
V. A. Dobrovol'skii, N. V. Lokot and J.-M. Strelcyn, Mikhail Nikolaevich Lagutinkskii (1871-1915): un mathématicien méconnu, Historia Math., 25 (1998), 245-264.
doi: 10.1006/hmat.1998.2194. |
[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), 211-246.
doi: 10.1007/BF01268661. |
[28] |
G. H. Halphen, Traité des Fonctions Elliptiques, Vol. I, II, Gauthier-Villars, Paris, 1888. |
[29] |
I. A. García, H. Giacomini and J. Giné, Generalized nonlinear superposition principles for polynomial planar vector fields, J. Lie Theory, 15 (2005), 89-104. |
[30] |
H. Giacomini , J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations, Rocky Mountain J. Math., 36 (2006), 457-485.
doi: 10.1216/rmjm/1181069462. |
[31] |
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. |
[32] |
J. P. Jouanolou, Equations de Pfaff Algébriques, Lectures Notes in Mathematics, 708, Springer-Verlag, New York/Berlin, 1979.
doi: 10.1007/BFb0063393. |
[33] |
I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. |
[34] |
T. Kimura, On Riemann's equations which are solvable by quadratures, Funkcialaj Ekvacioj, 12 (1969), 269-281. |
[35] |
J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3-43.
doi: 10.1016/S0747-7171(86)80010-4. |
[36] |
E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, Pure and Applied Mathematics, 54, New York - London, 1973. |
[37] |
A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. |
[38] |
J. Llibre and Ch. Pantazi, Polynomial differential systems having a given Darbouxian first integral, Bull. Sci. Math, 128 (2004), 775-788.
doi: 10.1016/j.bulsci.2004.04.001. |
[39] |
J. Llibre and Ch. Pantazi, Darboux theory of integrability for a class of nonautonomous vector fields, J. Math. Phys., 50 (2009), 102705, 19pp.
doi: 10.1063/1.3205450. |
[40] |
J. Llibre, Ch. Pantazi and S. Walcher, Morphisms and inverse problems for Darboux integrating factors, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1291-1302.
doi: 10.1017/S0308210511001430. |
[41] |
J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergod. Th. & Dynam. Sys., 31 (2011), 245-258.
doi: 10.1017/S0143385709000868. |
[42] |
J. Llibre and X. Zhang, Darboux theory of integrability in $\C^n$ taking into account the multiplicity, J. Diff. Equat., 246 (2009), 541-551.
doi: 10.1016/j.jde.2008.07.020. |
[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), 765-778.
doi: 10.1016/j.bulsci.2009.06.002. |
[44] |
B. Malgrange, Le groupoide de Galois d'un feuilletage, in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., 38 (2001), 465-501. |
[45] |
B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226.
doi: 10.1142/S0252959902000213. |
[46] |
J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resummation, in Computer Algebra and Differential Equations, (ed. E. Tournier), Comput. Math. Appl., Academic Press, London, (1990), 117-214. |
[47] |
J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, 179, Birkhäuser, Basel, 1999.
doi: 10.1007/978-3-0348-8718-2. |
[48] |
Ch. Pantazi, Inverse Problems of the Darboux Theory of Integrability for Planar Polynomial Differential Systems, PhD. thesis, Universitat Autonoma de Barcelona, 2004. |
[49] |
J. V. Pereira, Vector fields, invariant varieties and linear systems, Annales de l'institut Fourier, 51 (2001), 1385-1405.
doi: 10.5802/aif.1858. |
[50] |
J. V. Pereira, On the Poincaré problem for foliations of general type, Math. Ann. , 323 (2002), 217-226.
doi: 10.1007/s002080100277. |
[51] |
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition, Chapman and Hall, Boca Raton, 2003. |
[52] |
E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications, Inc., New York, 1960. |
[53] |
M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229.
doi: 10.1090/S0002-9947-1983-0704611-X. |
[54] |
M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-55750-7. |
[55] |
M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688.
doi: 10.1090/S0002-9947-1992-1062869-X. |
[56] |
M. F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations, Amer. J. Math., {103} (1981), 661-682.
doi: 10.2307/2374045. |
[57] |
F. Ulmer and J. A. Weil., Note on Kovacic's algorithm, J. Symb. Comp., 22 (1996), 179-200.
doi: 10.1006/jsco.1996.0047. |
[58] |
R. Vidunas, Differential equations of order two with one singular point, J. Symb. Comp., 28 (1999), 495-520.
doi: 10.1006/jsco.1999.0312. |
[59] |
J.-A. Weil, Constantes et Polynômes de Darboux en Algèbre Differentielle: Applications Aux Systèmes Différentiels Linéaires, PhD. Thesis, École Politechnique, 1995. |
[60] |
J. A. Weil, Recent algorithms for solving second-order differential equations, in Algorithms Seminar 2001-2002, (ed. F. Chyzak), INRIA, (2003), 43-46. |
[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, New York, 1962. |
[62] |
H. Zołądek, Polynomial Riccati equations with algebraic solutions, in Differential Galois theory (Bedlewo, 2001), Banach Center Publ., 58, Polish Acad. Sci., Warsaw, (2002), 219-231.
doi: 10.4064/bc58-0-17. |
show all references
References:
[1] |
P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics, PhD. Thesis, Universitat Politecnica de Catalunya, 2009, arXiv:0906.3532. |
[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, Berlin, 2010. |
[3] |
P. B. Acosta-Humánez, La teoría de Morales-Ramis y el algoritmo de Kovacic, Lecturas Matemáticas, 27 (2006), 21-56. |
[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), 279-297.
doi: 10.1137/080730329. |
[5] |
P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some Hamiltonians with rational potential, Discrete Continuous Dynam. Systems - B, 10 (2008), 265-293.
doi: 10.3934/dcdsb.2008.10.265. |
[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), 305-374.
doi: 10.1016/S0034-4877(11)60019-0. |
[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), Paper 043, 26 pp.
doi: 10.3842/SIGMA.2012.043. |
[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. |
[9] |
F. Baldassarri, On algebraic solutions of Lamé's differential equation, J. Diff. Equat., 41 (1981), 44-58.
doi: 10.1016/0022-0396(81)90052-8. |
[10] |
D. Blázquez-Sanz, Differential Galois Theory and Lie-Vessiot Systems, VDM Verlag Dr. Müller, Berlin, 2008. |
[11] |
D. Blázquez-Sanz and J.-J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in Differential algebra, complex analysis and orthogonal polynomials, (eds P.B. Acosta-Humánez and F. Marcellán), Contemp. Math., Amer. Math. Soc., Providence, 509 (2010), 1-58.
doi: 10.1090/conm/509. |
[12] |
D. Blázquez-Sanz and Ch. Pantazi, A note on the Darboux theory of integrability of non-autonomous polynomial differential systems, Nonlinearity, 25 (2012), 2615-2624.
doi: 10.1088/0951-7715/25/9/2615. |
[13] |
M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math., 140 (1994), 289-294.
doi: 10.2307/2118601. |
[14] |
G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres, Ann. Inst. Fourier, 56 (2006), 735-779.
doi: 10.5802/aif.2198. |
[15] |
D. Cerveau and A. Lins Neto, Holomorphic foliations in $\mathbbC \mathbbP (2)$ having an invariant algebraic curve, Ann. Inst. Fourier, 41 (1991), 883-903.
doi: 10.5802/aif.1278. |
[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), 9923-9949.
doi: 10.1088/0305-4470/37/42/007. |
[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), 217-229.
doi: 10.1017/S0956792503005114. |
[18] |
T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. |
[19] |
C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy.Soc. Edinburgh Sect. A, 124 (1994), 1209-1229.
doi: 10.1017/S0308210500030213. |
[20] |
C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Darboux integrating factors: Inverse problems, J. Diff. Equat., 250 (2011), 1-25.
doi: 10.1016/j.jde.2010.10.013. |
[21] |
C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves, Proc. Roy.Soc. Edinburgh Sect. A, 137 (2007), 1197-1226.
doi: 10.1017/S0308210506000400. |
[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), 287-302.
doi: 10.1017/S0308210507001175. |
[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), 2457-2476.
doi: 10.1088/0305-4470/35/10/310. |
[24] |
C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229 (2007), 63-117.
doi: 10.2140/pjm.2007.229.63. |
[25] |
G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200. |
[26] |
V. A. Dobrovol'skii, N. V. Lokot and J.-M. Strelcyn, Mikhail Nikolaevich Lagutinkskii (1871-1915): un mathématicien méconnu, Historia Math., 25 (1998), 245-264.
doi: 10.1006/hmat.1998.2194. |
[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), 211-246.
doi: 10.1007/BF01268661. |
[28] |
G. H. Halphen, Traité des Fonctions Elliptiques, Vol. I, II, Gauthier-Villars, Paris, 1888. |
[29] |
I. A. García, H. Giacomini and J. Giné, Generalized nonlinear superposition principles for polynomial planar vector fields, J. Lie Theory, 15 (2005), 89-104. |
[30] |
H. Giacomini , J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations, Rocky Mountain J. Math., 36 (2006), 457-485.
doi: 10.1216/rmjm/1181069462. |
[31] |
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. |
[32] |
J. P. Jouanolou, Equations de Pfaff Algébriques, Lectures Notes in Mathematics, 708, Springer-Verlag, New York/Berlin, 1979.
doi: 10.1007/BFb0063393. |
[33] |
I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. |
[34] |
T. Kimura, On Riemann's equations which are solvable by quadratures, Funkcialaj Ekvacioj, 12 (1969), 269-281. |
[35] |
J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3-43.
doi: 10.1016/S0747-7171(86)80010-4. |
[36] |
E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, Pure and Applied Mathematics, 54, New York - London, 1973. |
[37] |
A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. |
[38] |
J. Llibre and Ch. Pantazi, Polynomial differential systems having a given Darbouxian first integral, Bull. Sci. Math, 128 (2004), 775-788.
doi: 10.1016/j.bulsci.2004.04.001. |
[39] |
J. Llibre and Ch. Pantazi, Darboux theory of integrability for a class of nonautonomous vector fields, J. Math. Phys., 50 (2009), 102705, 19pp.
doi: 10.1063/1.3205450. |
[40] |
J. Llibre, Ch. Pantazi and S. Walcher, Morphisms and inverse problems for Darboux integrating factors, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1291-1302.
doi: 10.1017/S0308210511001430. |
[41] |
J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergod. Th. & Dynam. Sys., 31 (2011), 245-258.
doi: 10.1017/S0143385709000868. |
[42] |
J. Llibre and X. Zhang, Darboux theory of integrability in $\C^n$ taking into account the multiplicity, J. Diff. Equat., 246 (2009), 541-551.
doi: 10.1016/j.jde.2008.07.020. |
[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), 765-778.
doi: 10.1016/j.bulsci.2009.06.002. |
[44] |
B. Malgrange, Le groupoide de Galois d'un feuilletage, in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., 38 (2001), 465-501. |
[45] |
B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226.
doi: 10.1142/S0252959902000213. |
[46] |
J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resummation, in Computer Algebra and Differential Equations, (ed. E. Tournier), Comput. Math. Appl., Academic Press, London, (1990), 117-214. |
[47] |
J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, 179, Birkhäuser, Basel, 1999.
doi: 10.1007/978-3-0348-8718-2. |
[48] |
Ch. Pantazi, Inverse Problems of the Darboux Theory of Integrability for Planar Polynomial Differential Systems, PhD. thesis, Universitat Autonoma de Barcelona, 2004. |
[49] |
J. V. Pereira, Vector fields, invariant varieties and linear systems, Annales de l'institut Fourier, 51 (2001), 1385-1405.
doi: 10.5802/aif.1858. |
[50] |
J. V. Pereira, On the Poincaré problem for foliations of general type, Math. Ann. , 323 (2002), 217-226.
doi: 10.1007/s002080100277. |
[51] |
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Second Edition, Chapman and Hall, Boca Raton, 2003. |
[52] |
E. G. C. Poole, Introduction to the Theory of Linear Differential Equations, Dover Publications, Inc., New York, 1960. |
[53] |
M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229.
doi: 10.1090/S0002-9947-1983-0704611-X. |
[54] |
M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-55750-7. |
[55] |
M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688.
doi: 10.1090/S0002-9947-1992-1062869-X. |
[56] |
M. F. Singer, Liouvillian solutions of $n$th order homogeneous linear differential equations, Amer. J. Math., {103} (1981), 661-682.
doi: 10.2307/2374045. |
[57] |
F. Ulmer and J. A. Weil., Note on Kovacic's algorithm, J. Symb. Comp., 22 (1996), 179-200.
doi: 10.1006/jsco.1996.0047. |
[58] |
R. Vidunas, Differential equations of order two with one singular point, J. Symb. Comp., 28 (1999), 495-520.
doi: 10.1006/jsco.1999.0312. |
[59] |
J.-A. Weil, Constantes et Polynômes de Darboux en Algèbre Differentielle: Applications Aux Systèmes Différentiels Linéaires, PhD. Thesis, École Politechnique, 1995. |
[60] |
J. A. Weil, Recent algorithms for solving second-order differential equations, in Algorithms Seminar 2001-2002, (ed. F. Chyzak), INRIA, (2003), 43-46. |
[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, New York, 1962. |
[62] |
H. Zołądek, Polynomial Riccati equations with algebraic solutions, in Differential Galois theory (Bedlewo, 2001), Banach Center Publ., 58, Polish Acad. Sci., Warsaw, (2002), 219-231.
doi: 10.4064/bc58-0-17. |
[1] |
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