Abstract
In this paper, the Legendre polynomials approximations are studied for the numerical solutions of differential-algebraic equations (DAEs). An algorithm of the method constructed using Legendre polynomials approximations is developed to solve DAE systems. Two test problems are answered to illustrate the Legendre polynomials approximations, and then the obtained solutions are confronted with the exact solutions of the problems. As a first step, the power series of an allowed equation system is determined, in second step it is converted into Legendre polynomials approximations structure which allows an arbitrary order. Furthermore, to get numerical solution of DAEs with Legendre polynomials approximations, a Maple algorithm is implemented. Graphs of obtained solutions are sketched, and are created tables to compare the exact solutions by using Maple programming.