### Algebra + geometry differential equation solving.Algebra + geometry differential equation solving.

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 Author Winkler, Franz Source CiteSeerX Content type Text File Format PDF
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Rational General Solution ♦ Algebra Geometry Differential Equation ♦ Rational Parametrization ♦ Degree Bound ♦ Radical Differential Ideal ♦ Rational Solution ♦ Differential Algebra ♦ Algebraic Ode ♦ Autonomous Iff ♦ General Solution ♦ First Integral ♦ Positive Case ♦ Rational Invariant Algebraic Curve ♦ Tri-variate Polynomial ♦ Algebraic Surface ♦ Autonomous Ode ♦ Differential Field ♦ Bivariate Polynomial ♦ Non-autonomous Algebraic Ode ♦ Problem Rational ♦ Autonomous Case ♦ Order Classification ♦ Differential Polynomial ♦ Aodes Reference J.f. Ritt ♦ Algebraic Ordinary Differential Equation ♦ Plane Algebraic Curve Abstract Consider an autonomous ODE of the form F(y,y ′ ) = 0, where F is a bivariate polynomial. We can think of F as defining a plane algebraic curve. If this curve admits a rational parametrization, then we can determine whether the ODE has a rational general solution. Feng and Gao have described an algorithm for this problem, based on degree bounds for such parametrizations by Sendra and Winkler. Here we extend this investigation to the case of non-autonomous algebraic ODEs of the form F(x,y,y ′ ) = 0. The tri-variate polynomial F defines an algebraic surface, which we assume to admit a rational parametrization. Based on such a parametrization and on knowledge about a degree bound for rational solutions, we can determine the existence of a rational general solution, and, in the positive case, also compute one. This method depends crucially on the determination of rational invariant algebraic curves. We also relate rational general solutions of algebraic ODEs to rational first integrals. Outline The problem Rational parametrizations The autonomous case The general (non-autonomous) case Solving the associated system Rational general solutions Generalization to higher order Classification of AODEs References J.F. Ritt, Differential Algebra (1950) E. Hubert, The general solution of an ODE, Proc. ISSAC 1996 The problem An algebraic ordinary differential equation (AODE) is given by F(x,y,y ′,...,y (n) ) = 0, where F is a differential polynomial in K[x]{y} with K being a differential field and the derivation ′ being d dx. Such an AODE is autonomous iff F does not depend on x; i.e., F ∈ K{y}. The radical differential ideal {F} can be decomposed Educational Role Student ♦ Teacher Age Range above 22 year Educational Use Research Education Level UG and PG ♦ Career/Technical Study