Frobenius Method: General Case of Regular Singular Point

This lecture presents a constructive proof of Fuchs' Theorem, focusing on applying the series substitution method to the general case of an ordinary or regular singular point xo. It explores scenarios at xo, considering a 2nd order ODE with power series expressions for p(x) and q(x). The lecture delves into the nature of roots S1 and S2, the indicial equation, and deriving the second solution in different cases. It illustrates the method through examples and emphasizes the importance of ordinary points. The Frobenius series solutions and their properties are thoroughly discussed, providing insights into linearly independent solutions and the Wronskian. The lecture concludes by showcasing the application of the method to the harmonic oscillator ODE.