A Unified Recurrence Framework for Power-Series and Frobenius Solutions of Second-Order Linear ODEs

This paper presents a unified coefficient-recurrence framework for obtaining series solutions of second-order linear ordinary differential equations with variable coefficients. The standard power-series method, applicable at ordinary points, and the Frobenius method, applicable at regular singular points, are treated as special cases of a common series ansatz $y(x)=(x-x_0)^r\sum_{n=0}^{\infty}a_n(x-x_0)^n$, where $r=0$ recovers the ordinary case and $r$ is determined by an indicial equation in the singular case. General recurrence relations for analytic coefficient functions are derived, conditions under which series terminate to yield polynomial solutions are established, and resonance in the integer-difference root case is characterized. Carefully selected examples illustrate the framework, including polynomial solutions, fractional exponents, resonance with logarithmic terms, and the Bessel equation. The analysis provides a systematic structural comparison of the two methods in terms of recurrence order, termination, resonance, and computational complexity.