the Bessel equation
/ BESS-ul /
Whenever a problem has circular or cylindrical symmetry — the vibrations of a circular drumhead, the temperature in a round pipe, the diffraction of light through a circular aperture, waves in a cylindrical waveguide — separating variables in the radial direction produces this one equation. The Bessel equation is the cylindrical-symmetry counterpart of the Legendre equation, and it is just as ubiquitous in physics and engineering.
It reads x^2 y'' + x y' + (x^2 - nu^2) y = 0, where the parameter nu (the 'order', often a non-negative real number) is fixed by the problem, and x is typically a scaled radius. The point x = 0 — the centre of the disk or pipe — is a regular singular point, so you cannot use an ordinary power series there; you must use the method of Frobenius. Its indicial equation is r^2 - nu^2 = 0, with roots r = +nu and r = -nu, and the resulting Frobenius series defines the Bessel functions. The radius enters through the x^2 term, which makes the equation oscillatory for large x, so its solutions wobble up and down like damped sines and cosines but with slowly changing amplitude and spacing.
The Bessel equation is the textbook home of the full method of Frobenius, because the difference of its indicial roots is 2nu, which lands you in every one of the three cases as nu varies — non-integer (clean), half-integer (clean but special), and integer (where the dreaded logarithm appears). That makes it not just physically central but the canonical example for learning how singular-point series solutions really behave.
A vibrating circular drum of radius a obeys, in its radial part, the Bessel equation of order 0. The allowed vibration frequencies are set by requiring the solution J_0 to vanish at the rim r = a — so the drum's overtones are spaced by the zeros of J_0, which (unlike a string's) are not evenly spaced, which is why a drum sounds inharmonic.
The boundary condition picks out the zeros of a Bessel function as the allowed modes — and those zeros are unevenly spaced, unlike a string's harmonics.
Despite the x^2 in the equation, the constant nu is a fixed parameter, not a power to be solved for; the actual exponents that emerge are +-nu, from the indicial equation, not nu itself.