The Bessel Differential Equation is a second-order linear differential equation that arises in various physical problems, particularly in cylindrical coordinates. It is typically expressed in the form x^2 y'' + x y' + (x^2 - n^2) y = 0 , where n is a constant. Solutions to this equation are known as Bessel functions, which are important in fields such as acoustics, electromagnetism, and heat conduction. Bessel functions come in two main types: Bessel functions of the first kind and Bessel functions of the second kind. The first kind, denoted as J_n(x) , is finite at the origin for non-negative integer n , while the second kind, denoted as Y_n(x) , has a singularity at the origin. These functions are essential for solving problems with circular or cylindrical symmetry.