Mathieu Equation
The canonical form for Mathieu's differential equation is
The Mathieu equation is a Hill equation with only 1 harmonic mode. Closely related is Mathieu's modified differential equation
which follows on substitution .
The substitution transforms Mathieu's equation to the algebraic form
This has two regular singularities at and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions.
Mathieu's differential equations arise as models in many contexts, including the stability of railroad rails as trains drive over them, seasonally forced population dynamics, the four-dimensional wave equation, and the Floquet theory of the stability of limit cycles.
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