Power series solutions of second-order ODEs near ordinary points via recurrence relations and Mathematica/Python

This article presents a simplified approach to deriving recurrence relations for power series solutions of second-order ordinary differential equations (ODEs) near ordinary points. Instead of the traditional symbolically intensive procedure of substituting a series into the differential equation, aligning powers, and shifting indices, we employ General Recurrence Formulas (RF), which allow recurrence relations to be identified almost by inspection. This reduces algebraic complexity and minimises computation errors. To support teaching and self-study, the method is combined with modern computational tools, including Mathematica and Python. These environments automate the computation of power series coefficients and facilitate the rapid creation of truncated solutions, helping students focus on conceptual understanding rather than lengthy symbolic manipulation. Fully functional Mathematica and Python code and several worked examples illustrate the method and demonstrate its advantages in clarity, efficiency, and accessibility for learning environments.