Kassahun WorkalemahuGebrye T/Mariam2025-10-082025-10-082024-09https://etd.aau.edu.et/handle/123456789/7490Dif erential equations play an important role in solving real life problems, and we knowthatsolving dif erential equation is not simple in general. In this thesis we take the Bessel dif erentialequation and discuss the methods to solve the dif erential equation using Frobenius method. Webegin by finding Bessel functions Jn(x) and Yn(x) which are two solutions of Bessel dif erentialequation named as Bessel functions of first and second kind will also be described, whennisaninteger. For our purpose we consider some properties and proofs of recurrence relations, generating functions and special functions. Finally we discuss about the vibration of circularmembrane and its solutions.en-USBessel Differential EquationFrobenius MethodGenerating FunctionsBessel Functions and ApplicationThesis