on the conditions of similar analytical solutions of homotopy perturbation, taylor series and differential transformation methods

The developments of approximate analytical solutions to nonlinear differential equations have been achieved through the use of various approximate analytical and semi-analytical methods. these methods provide different analytical expressions which give difference values for the same input data and variables. however, under some certain conditions, the methods provide similar analytical expressions, thereby give the same values for the same input data and variables. therefore, in this work, the conditions of similar analytical solutions by homotopy perturbation, differential transformation and taylor series methods for linear and nonlinear differential equations are investigated. from the analysis, it is established that if some specific values or functions are assigned to the auxiliary parameters in the homotopy perturbation method, the approximate analytical solutions provided by homotopy perturbation method is entirely similar to the approximate analytical solutions given by differential transformation and taylor series methods. also, it is found that the results of taylor series method when expansion is at the center, is exactly the same to the results of homotopy perturbation and differential transformation methods. it is hoped that this work will great assist and enhance the understanding of mathematical solutions providers and enthusiasts as it provides better insight into finding analytical solutions to linear and nonlinear differential equations.