# Mathieu equation

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The following ordinary differential equation with real coefficients:

It was introduced by E. Mathieu [1] in the investigation of the oscillations of an elliptic membrane; it is a particular case of a Hill equation.

A fundamental system of solutions of the Mathieu equation has the form

 (*)

for , an integer, where is a -periodic function and the Lyapunov characteristic exponent is either real or purely imaginary. For one of the solutions grows unboundedly, whereas the other tends to zero as (instability zones in the plane of the parameters ); for these solutions are both bounded (stability zones). On the boundary of these zones (the case excluded in (*)) one of the functions of the fundamental system of solutions is either -periodic or -periodic (the latter is called a Mathieu function, cf. Mathieu functions), while the second is obtained from the first through multiplication by . The instability zones have the form of curvilinear triangles with vertices at the points , , (see [2], [4]).

The Mathieu equation is known also in a different form (see [3]).

#### References

 [1] E. Mathieu, "Course de physique mathématique" , Paris (1873) [2] M.J.O. Strett, "Lamésche-, Mathieusche- und verwandte Funktionen in Physik und Technik" , Springer (1932) [3] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971) [4] V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients and their applications" , 1–2 , Wiley (1975) (Translated from Russian)