A professional semi-analytical method to construct series solution to singular and nonsingular differential equations

This paper introduces an attractive novel technique named the limit residual function method for finding series solutions of differential equations. The concepts of residual function and the limit at zero are the main tools of this approach. The new method has the advantage of quickly determining the coefficients of the series solution and the limited calculations required compared to other methods. We demonstrate the effectiveness and versatility of this method by applying it to various types of differential equations, including both linear and non-linear, some singular and others non-singular. To authenticate the accuracy and reliability of the method, we also present numerical simulations. The proposed method can find exact solutions when a pattern can be found in the series solution, and it can provide approximate solutions when no such pattern exists. 32A05; 37C83; 34A45; 46N20.