Abstract
We apply Kovacic's algorithm, a tool that is developed from differential Galois theory, to discuss the existence of Liouvillian solutions of Whittaker-Ince equation, ellipsoidal wave equation and the Picard-Fuchs equation of a K3 surface. We determine the necessary and sufficient conditions of having Liouvillian solutions for Whittaker-Ince equation when a parameter vanishes; as well as a sufficient condition of having Liouvillian solution when this parameter is non-zero. On the other hand, we prove that ellipsoidal wave equation has no Liouvillian solution. We generalize a Picard-Fuchs equation for certain K3 surface and show that a particular case of this Picard-Fuchs equation cannot have any Liouvillian solutions.
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