2019/06/10 by V. León, León, V., Bruno Scárdua +2
Physics and Astronomy · Mathematics · Computer Science · #Nonlinear Waves and Solitons #Advanced Topics in Algebra #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1906.04277
We study second order and third order linear differential equations with\nanalytic coefficients under the viewpoint of finding formal solutions and\nstudying their convergence. We address some untouched aspects of Frobenius\nmethods for second order as the convergence of formal solutions and the\nexistence of Liouvillian solutions. A characterization of regular singularities\nis given in terms of the space of solutions. An analytic classification of such\nlinear homogeneous ODEs is obtained. This is done by associating to such an ODE\na Riccati differential equation and therefore a global holonomy group. This\ngroup is a computable group of Moebius maps. These techniques apply to\nclassical equations as Bessel and Legendre equations. In the second part of\nthis work we study third order equations. We prove a theorem similar to\nclassical Frobenius theorem, which describes all the possible cases and\nsolutions to this type of ODE. Once armed with this we pass to investigate the\nexistence of solutions in the non-homogeneous case and also the existence of a\nconvergence theorem in the same line as done for second order above. Our\nresults are concrete and (computationally) constructive and are aimed to shed a\nnew light in this important, useful and attractive field of science.\n