"Strong Rational General Solutions of AODEs using Optimal Curve Parametrization"
, Serie Research Institute for Symbolic Computation, Lukas Weigert, 4-2020
Strong Rational General Solutions of AODEs using Optimal Curve Parametrization
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The aim of this thesis is to implement an algorithm that finds rational solutions of first-order algebraic ordinary differential equations (AODEs). We are interested in the general solution that depends on a transcendental constant. To tackle this problem, a geometric approach is used. We neglect the differential aspect and consider the derivative as new variable. This leads to an algebraic equation. So we consider the AODE as an algebraic curve, in which the coefficients are rational functions. We have to compute the rational parametrization of the obtained curve. Since we look for rational solutions, we also require the coefficients of the parametrization to be rational. Every curve over the field of rational functions admits such a parametrization. Therefore, the key notion is optimal parametrization. For parametrizing over the rational numbers, there are already implementations available. But these are not applicable for our problem, since they require field extensions. So we have to construct a new implementation. Our goal is to decide whether the AODE has a rational general solution and in the affirmative case compute it. To do so, we have to modify the problem and search for solutions where also the arbitrary constant appears rationally. Such a solution is called strong rational general solution. Thus, we achieve a decision algorithm.