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: the first kind, denoted as J_n(x) , and the second kind, denoted as Y_n(x) . These functions exhibit oscillatory behavior and are used to model waveforms and vibrations in cylindrical systems. The Bessel differential equation and its solutions are essential in engineering and physics, particularly in problems involving circular or cylindrical symmetry.