Superiority of Legendre Polynomials to Chebyshev Polynomial in Solving Ordinary Differential Equation

In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev polynomial T*n(X), defined with the help of hypergeometric series T*n(x) =F ( -n, n, ½ ;X) and later choosing Legendre polynomial P*n (x) define by the series P*n (x) = F ( -n, n+1, 1;X); with the help of an auxiliary set of Canonical polynomials Qk in order to find the superiority between the two polynomials. Numerical examples are given which show the superiority of Legendre polynomials to Chebyshev polynomials.