Approximate solution to linear complex differential equation by a new approximate approach

The purpose of this study is to give a Taylor polynomial approximation for the solution of high-order linear complex differential equations with variable coefficients and mixed conditions about any point. For this purpose, a matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the complex differential equation and then substituting their matrix forms into the given equation. Hence the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. Also, examples that illustrate the pertinent features of the method are presented and the results are discussed.