Bessel’s equation

Bessel's equation is a differential equation that arises in the solution of many physical problems, particularly those involving cylindrical symmetry. It is a second-order linear differential equation of the form:1

[
x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0
]

The solutions to Bessel's equation are known as Bessel functions, denoted by (J_n(x)) and (Y_n(x)). These functions were first systematically derived by the German astronomer Friedrich Wilhelm Bessel in the early 19th century. Bessel functions appear in problems related to heat flow, wave propagation, and electromagnetism in cylindrical systems. For example, they are used to describe the vibrations of a stretched membrane or the oscillations of a chain.1234