Möbius transformations on algebraic ODEs of order one and algebraic general solutions of the autonomous equivalence classes

Abstract

We define a birational equivalence of algebraic ordinary differential equations of order one and study two invariant properties related to algebraic general solutions of the differential equations and the notion of differential total degree. We use this equivalence and its invariants to deduce a new degree bound for algebraic general solutions of algebraic ordinary differential equations of order one in the equivalence classes of autonomous ones. The degree bound can be used to decide the existence of algebraic general solutions of the autonomous class of algebraic ordinary differential equations of order one. On the other hand, we propose an algorithm for computing an algebraic general solution of the algebraic ordinary differential equations of order one in the autonomous classes.