Bessel's Equation

Differential Equations · Axiom Academy

From a vibrating drumhead to a single ODE — deriving Bessel's equation, solving it by Frobenius series, and reading off the physical modes. 1. From the Wave Equation to Bessel's Equation Start with the 2-D wave equation for a vibrating membrane, written in polar coordinates so the circular symmetry of a drumhead is built in from the start. The wave equation in polar coordinates Separating variables with splits this into three ordinary equations. The angular piece forces to be -periodic, which pins its separation constant to an integer n ; the radial piece is left as: 2. Solving by Frobenius Series Bessel's equation of order has a regular singular point at x=0 , so we solve it with a Frobenius series . Matching every other power of x gives the recurrence — odd coefficients vanish, and the even ones telescope into a single closed form built entirely from factorials and the Gamma function : Bessel function of the first kind (the solution bounded at x=0 ) Bessel function of the second kind : the independent solution built from (Neumann function). When is an integer, the two indicial roots differ by an integer, and the standard Frobenius theory then forces to pick up a logarithmic term so the two solutions stay independent. Either way, Y_ (x) - as x 0^+ . 3. Application: The Vibrating Drumhead

This is the written version of the interactive lesson above. See the full Differential Equations course.