General Solutions of Some Complex Third-order Differential Equations

Abstract

According to the Nevanlinna theory, many researches have undertaken the behaviors of meromorphic solutions of complex ordinary differential equations (ODEs). Most of these researches have concentrated on the value distribution and growth of meromorphic solutions of ODEs. However, the existence of a meromorphic general solution is often used as a way to identify equations that are integrable. Especially, the existence of global meromorphic solutions of differential equation with entire coefficient can be settled, resulting in the characterization of Schwarzian derivatives. This is concerning with the linearly independent solutions of linear differential equations . The purpose of this present paper is to find explicit solutions of differential equation in terms of finite combinations of known functions, that is, we use local series methods and reduction of order to solve all linearly independent solutions of some third-order ODEs with entire coefficient in the neighborhood of z0.