Frobenius Method: Illustration around an Ordinary Point

This lecture introduces the Frobenius method for solving ordinary differential equations around an ordinary point. The instructor explains how to derive a power series solution and determine the exponent and coefficient ratios. The Fuchs Theorem conditions for the existence of solutions are discussed, distinguishing between ordinary, regular singular, and irregular singular points. Examples illustrate the application of the method to the harmonic oscillator equation, emphasizing the importance of choosing the point of expansion. The process of finding solutions through series substitution is detailed, including deriving the indicial equation and recurrence relations. The lecture concludes by demonstrating the derivation of two linearly independent solutions using the Frobenius series, leading to the general solution of the harmonic oscillator equation.