June  2021, 29(2): 2101-2127. doi: 10.3934/era.2020107

A geometric-analytic study of linear differential equations of order two

1. 

Instituto Latino-Americano de Ciências da Vida e da Natureza, Centro Interdisciplinar de Ciências da Natureza, Universidade Federal da Integração Latino-Americana, Parque tecnológico de Itaipu, Foz do Iguaçu-PR, 85867-970, Brazil

2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro-RJ, 21945-970, Brazil

* Corresponding author: Bruno Scárdua

Received  February 2020 Revised  August 2020 Published  June 2021 Early access  October 2020

We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a global holonomy group. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.

Citation: Víctor León, Bruno Scárdua. A geometric-analytic study of linear differential equations of order two. Electronic Research Archive, 2021, 29 (2) : 2101-2127. doi: 10.3934/era.2020107
References:
[1]

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag, Berlin-Göttingen-Heidelberg; Academic Press Inc., New York 1957.  Google Scholar

[2]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10$^th$ edition, John Wiley & Sons, New York. 2012. Google Scholar

[3]

C. Camacho and B. Azevedo Scárdua, Holomorphic foliations with Liouvillian first integrals, Ergod. Theory Dyn. Syst., 21 (2001), 717-756.  doi: 10.1017/S0143385701001353.  Google Scholar

[4]

C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Progress in Math. Birkhäusser, Boston, Basel and Stuttgart, 1985. doi: 10.1007/978-1-4612-5292-4.  Google Scholar

[5]

D. Cerveau and J.-F. Mattei, Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982.  Google Scholar

[6]

E. A. Coddington, An Introduction to Ordinary Differential Equations, Dover Publications, New York, 1989. Google Scholar

[7]

H. M. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Mathematics, 71. Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[8]

G. Frobenius, Ueber die Integration der linearen Differentialgleichungen durch Reihen, J. Reine Angew. Math., 76 (1873), 214-235.  doi: 10.1515/crll.1873.76.214.  Google Scholar

[9]

C. Godbillon, Feuilletages. Études géométriques, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991.  Google Scholar

[10]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994. doi: 10.1002/9781118032527.  Google Scholar

[11]

G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math., 8 (1886), 1-36.  doi: 10.1007/BF02417081.  Google Scholar

[12]

J. Kepler, Astronomia Nova, 1609, edited in J. Kepler, Gesammelte Werke, vol Ⅲ, 2$^nd$ edition, C.H. Beck, München, 1990. Google Scholar

[13]

A. Lins Neto and B. Scárdua, Complex Algebraic Foliations, Expositions in Mathematics, vol. 67, Walter de Gruyter GmbH, Berlin/Boston, 2020. Google Scholar

[14]

B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.   Google Scholar

[15]

B. Malgrange, Frobenius avec singularités. Ⅱ. Le cas général, Invent. Math., 39 (1977), 67-89.  doi: 10.1007/BF01695953.  Google Scholar

[16]

J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup., 13 (1980), 469-523.   Google Scholar

[17]

I. Newton, The Mathematical Principles of Natural Philosophy, Bernard Cohen Dawsons of Pall Mall, London, 1968.  Google Scholar

[18]

P. Painlevé, Leçcons sur la Théorie Analytique des Équations Différentielles, Librairie Scientifique A. Hermann, Paris, 1897. Google Scholar

[19]

F. Reis, Methods from Holomorphic Foliations in Differential Equations, Ph. D thesis, IM-UFRJ in Rio de Janeiro, 2019. Google Scholar

[20]

M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. Math., 65 (1976), 485-492.  doi: 10.2140/pjm.1976.65.485.  Google Scholar

[21]

F. Santos and B. Scárdua, Construction of vector fields and Ricatti foliations associated to groups of projective automorphism, Conform. Geom. Dyn., 14 (2010), 154-166.  doi: 10.1090/S1088-4173-2010-00208-0.  Google Scholar

[22]

B. A. Scárdua, Transversely affine and transversely projective holomorphic foliations, Ann. Sci. École Norm. Sup., 30 (1997), 169-204.  doi: 10.1016/S0012-9593(97)89918-1.  Google Scholar

[23]

B. Scárdua, Differential algebra and Liouvillian first integrals of foliations, J. Pure Appl. Algebra, 215 (2011), 764-788.  doi: 10.1016/j.jpaa.2010.06.023.  Google Scholar

[24]

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.  Google Scholar

show all references

References:
[1]

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag, Berlin-Göttingen-Heidelberg; Academic Press Inc., New York 1957.  Google Scholar

[2]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10$^th$ edition, John Wiley & Sons, New York. 2012. Google Scholar

[3]

C. Camacho and B. Azevedo Scárdua, Holomorphic foliations with Liouvillian first integrals, Ergod. Theory Dyn. Syst., 21 (2001), 717-756.  doi: 10.1017/S0143385701001353.  Google Scholar

[4]

C. Camacho and A. Lins Neto, Geometric Theory of Foliations, Progress in Math. Birkhäusser, Boston, Basel and Stuttgart, 1985. doi: 10.1007/978-1-4612-5292-4.  Google Scholar

[5]

D. Cerveau and J.-F. Mattei, Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982.  Google Scholar

[6]

E. A. Coddington, An Introduction to Ordinary Differential Equations, Dover Publications, New York, 1989. Google Scholar

[7]

H. M. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Mathematics, 71. Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[8]

G. Frobenius, Ueber die Integration der linearen Differentialgleichungen durch Reihen, J. Reine Angew. Math., 76 (1873), 214-235.  doi: 10.1515/crll.1873.76.214.  Google Scholar

[9]

C. Godbillon, Feuilletages. Études géométriques, Progress in Mathematics, vol. 98, Birkhäuser Verlag, Basel, 1991.  Google Scholar

[10]

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994. doi: 10.1002/9781118032527.  Google Scholar

[11]

G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math., 8 (1886), 1-36.  doi: 10.1007/BF02417081.  Google Scholar

[12]

J. Kepler, Astronomia Nova, 1609, edited in J. Kepler, Gesammelte Werke, vol Ⅲ, 2$^nd$ edition, C.H. Beck, München, 1990. Google Scholar

[13]

A. Lins Neto and B. Scárdua, Complex Algebraic Foliations, Expositions in Mathematics, vol. 67, Walter de Gruyter GmbH, Berlin/Boston, 2020. Google Scholar

[14]

B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.   Google Scholar

[15]

B. Malgrange, Frobenius avec singularités. Ⅱ. Le cas général, Invent. Math., 39 (1977), 67-89.  doi: 10.1007/BF01695953.  Google Scholar

[16]

J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup., 13 (1980), 469-523.   Google Scholar

[17]

I. Newton, The Mathematical Principles of Natural Philosophy, Bernard Cohen Dawsons of Pall Mall, London, 1968.  Google Scholar

[18]

P. Painlevé, Leçcons sur la Théorie Analytique des Équations Différentielles, Librairie Scientifique A. Hermann, Paris, 1897. Google Scholar

[19]

F. Reis, Methods from Holomorphic Foliations in Differential Equations, Ph. D thesis, IM-UFRJ in Rio de Janeiro, 2019. Google Scholar

[20]

M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. Math., 65 (1976), 485-492.  doi: 10.2140/pjm.1976.65.485.  Google Scholar

[21]

F. Santos and B. Scárdua, Construction of vector fields and Ricatti foliations associated to groups of projective automorphism, Conform. Geom. Dyn., 14 (2010), 154-166.  doi: 10.1090/S1088-4173-2010-00208-0.  Google Scholar

[22]

B. A. Scárdua, Transversely affine and transversely projective holomorphic foliations, Ann. Sci. École Norm. Sup., 30 (1997), 169-204.  doi: 10.1016/S0012-9593(97)89918-1.  Google Scholar

[23]

B. Scárdua, Differential algebra and Liouvillian first integrals of foliations, J. Pure Appl. Algebra, 215 (2011), 764-788.  doi: 10.1016/j.jpaa.2010.06.023.  Google Scholar

[24]

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.  Google Scholar

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