This paper investigates a highly efficient method that depends on the Tau method for solving initial and boundary value problems. The second derivatives of Legendre polynomials (SDLPs) have been used as novel basis functions. A linearization relation for the presented basis functions has been introduced and proved to avoid any issues arising during tau’s integration, especially for the nonlinear problems. Consequently, some essential integrations have been determined. Moreover, we used those relations to construct explicit forms for approximating the solutions of Lane-Emden and the Recatti equations. In addition, the presented strategy’s converge and error analysis are discussed carefully and in-depth. Finally, the mentioned IVPs have been solved via the proposed method. The results have been compared with the others’ methods, which showed our technique’s accuracy, efficiency, and stability.