A Systematic Method for Deriving Recurrence Relation in Second-Order Linear Ordinary Differential Equations

We study power-series solutions of homogeneous second-order linear ordinary differential equations whose coefficients are quadratic polynomials. The main contribution is a compact matrix formulation that yields the coefficient recurrence relation in a direct and transparent way, avoiding the usual bookkeeping with multiple summations and index shifts. The method is well suited for classroom exposition and explicit computation. The same framework also supports a reverse perspective: starting from a prescribed series behavior, one may construct differential equations for which that behavior arises as a solution.