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Rational Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One
Nguyen Tri Dat
,
Ngo Lam Xuan Chau
Abstract
We give a necessary and sufficient condition for an autonomous first-order AODE to have a rational liouvillian solution. We also give an algorithm to compute a rational liouvillian general solution if it exists. The algorithm is based on the rational parametrization of the corresponding algebraic curve of the first-order autonomous AODE and the existence of a rational liouvillian element over $\mathbb C$. When the corresponding algebraic curve is rational, this method covers the known cases of rational solutions and radical solutions.