23,839 research outputs found

    Rational general solutions of algebraic ordinary differential equations

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    We give a necessary and sufficient condition for an alge-braic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to com-pute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants

    Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-Geometric Dimension One

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    The final journal version of this paper appears in A. Lastra, J. R. Sendra, L. X. C. Ngô and F. Winkler\ud (2014). Rational General Solutions of Systems of Autonomous Ordinary Differential Equations of Algebro-\ud Geometric Dimension One. Publ. Math. Debrecen Publ. Math. Debrecen 2015 / 86 / 1-2 49–69. DOI:\ud 10.5486/PMD.2015.6032 and it is available at http://dx.doi.org/10.5486/PMD.2015.6032An algebro-geometric method for determining the rational solvability\ud of autonomous algebraic ordinary differential equations is extended from single equations\ud of order 1 to systems of equations of arbitrary order but dimension 1 in the algebrogeometric\ud sense. We provide necessary conditions, for the existence of rational solutions,\ud on the degree and on the structure at infinity of the associated algebraic curve. Furthermore,\ud from a rational parametrization of a planar projection of the corresponding\ud space curve one deduces, either by derivation or by lifting the planar parametrization,\ud the existence and actual computation of all rational solutions if they exist. Moreover, if\ud the differential polynomials are defined over the rational numbers, we can express the\ud rational solutions over the same field of coefficients.Vietnam Institute for Advanced Study in Mathematics (VIASM

    Integrability of planar polynomial differential systems through linear differential equations

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    In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.Comment: 24 pages, no figure

    On Symbolic Solutions of Algebraic Partial Differential Equations

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    The final version of this paper appears in Grasegger G., Lastra A., Sendra J.R. and\ud Winkler F. (2014). On symbolic solutions of algebraic partial differential equations, Proc.\ud CASC 2014 SpringerVerlag LNCS 8660 pp. 111-120. DOI 10.1007/978-3-319-10515-4_9\ud and it is available at at Springer via http://DOI 10.1007/978-3-319-10515-4_9In this paper we present a general procedure for solving rst-order autonomous\ud algebraic partial di erential equations in two independent variables.\ud The method uses proper rational parametrizations of algebraic surfaces\ud and generalizes a similar procedure for rst-order autonomous ordinary\ud di erential equations. We will demonstrate in examples that, depending on\ud certain steps in the procedure, rational, radical or even non-algebraic solutions\ud can be found. Solutions computed by the procedure will depend on\ud two arbitrary independent constants

    A computer algebra approach to rational general solutions of algebraic ordinary differential equations

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    In this thesis, I approach to algebraic ODEs from Differential Algebra's point of view. I look for rational solutions of AODE, I present an algebro-geometric method to decide the existence of rational solutions of a first-order algebraic ODE and if they exist an algorithm to compute them. This method depends heavily on rational parametrizations, in particular for autonomous equations on the parametrization of algebraic curves, and for non-autonomous equations on the parametrization of algebraic surfaces. In the last case, I prove the correspondence between rational solutions of a parametrizable algebraic ODE and rational solutions of a first-order linear autonomous differential system of two equations in two variables. I provide an algorithm to compute rational solutions of such system based on its invariant algebraic curves. I also study a group of affine transformations which preserves the rational solvability, in order to reduce, when possible, an algebraic ODE to an easier one. Moreover I present the results of the implementation of all these algorithms in two computer algebra system: CoCoA and Singular
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