This course explores advanced topics in ordinary and fractional differential equations . Students study power series solutions near singular points, the Frobenius method, Gamma and Beta functions, and fractional factorials. The course then introduces fractional calculus with Caputo derivatives, Mittag?Leffler functions, fractional sine and cosine functions, and Laplace transform methods for fractional IVPs. Modern series approaches such as the Residual Power Series Method (RPSM) and its fractional version (FRPSM) are also presented.