SERIES SOLUTION METHOD FOR SECOND ORDER LINEAR DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENT.

ABSTRACT This research focuses on the application of the power series method to solve second-order linear differential equations with constant coefficients, particularly in cases where traditional methods such as the characteristic equation or the method of undetermined coefficients may be insufficient. The study investigates the theoretical foundation of the power series method, deriving recurrence relations that enable iterative solutions. Emphasis is placed on the method’s ability to generate special functions, such as Bessel and Legendre functions, which are essential in mathematical physics and engineering. Through analytical demonstration and comparison with classical methods, the research highlights the versatility and effectiveness of the power series approach. Real-life applications in quantum mechanics, heat conduction, and wave propagation are also examined to demonstrate its practical relevance. The study concludes that the power series method remains a vital analytical tool for solving complex differential equations.