Generalized trial equation method and its applications to Duffing and Poisson-Boltzmann equations

The trial equation method, which was proposed by Cheng-Shi Liu, is a very powerful method for solving nonlinear differential equations. After the original trial method, some modified versions of the trial equation method were introduced and applied to some famous nonlinear differential equations. Although each modified trial equation method provides a different perspective, they have some weaknesses according to the given differential equations. This is the main reason for introducing modified trial equation methods. This study aims to define a general representation of trial methods for solving nonlinear differential equations. The generalized trial equation method consists of the simple trial equation method, irrational trial method, and extended trial equation method as a common coverage. A suitable trial equation can also be structured according to the given nonlinear differential equations. To demonstrate the applicability of the generalized trial equation method, the solutions of the Duffing equation and Poisson Boltzmann equation are examined and new solutions of these equations are obtained based on some nonlinear functions that have not been considered before within the trial equation methods.