In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation where
[1] It is named after George William Hill, who introduced it in 1886.
, the Hill equation can be rewritten using the Fourier series of
: Important special cases of Hill's equation include the Mathieu equation (in which only the terms corresponding to n = 0, 1 are included) and the Meissner equation.
Depending on the exact shape of
, solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially.
[3] The precise form of the solutions to Hill's equation is described by Floquet theory.
Solutions can also be written in terms of Hill determinants.
[1] Aside from its original application to lunar stability,[2] the Hill equation appears in many settings including in modeling of a quadrupole mass spectrometer,[4] as the one-dimensional Schrödinger equation of an electron in a crystal,[5] quantum optics of two-level systems, accelerator physics and electromagnetic structures that are periodic in space[6] and/or in time.