The Mathieu equation is a second-order homogeneous linear differential equation and appears in several different situations in Physics: electromagnetic or elastic wave equations with elliptical boundary conditions as in waveguides or resonators, the motion of particles in alternated-gradient focussing or electromagnetic traps, the inverted pendulum, parametric oscillators, the motion of a quantum particle in a periodic potential, the eigenfunctions of the quantum pendulum, are just few examples.

Their solutions, known as Mathieu functions, were first discussed by Mathieu in 1868 in the context of the free oscillations of an elliptic membrane:

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