This paper investigates Lucas polynomials through the spectral tau method to construct an approximate semi-analytic technique for solving both linear and nonlinear ordinary differential equations. Two computational schemes are developed using operational matrices of Lucas polynomial derivatives to transform the governing equations into matrix equations. The Lane–Emden equations are considered as the main applications due to their importance in astrophysics and mathematical physics. Numerical and graphical results demonstrate the high accuracy, convergence, and computational efficiency of the proposed algorithms when compared with existing numerical techniques. The study confirms the effectiveness of Lucas polynomial operational matrices in solving a wide range of boundary value and initial value problems.