Meissner equation

The Meissner equation is a linear ordinary differential equation that is a special case of Hill's equation with the periodic function given as a square wave. There are many ways to write the Meissner equation. One is as

or

where

and ${\displaystyle H_{c}(t)}$ is the Heaviside function shifted to ${\displaystyle c}$. Another version is

The Meissner equation was first studied as a toy problem for certain resonance problems. It is also useful for understand resonance problems in evolutionary biology.

Because the time-dependence is piecewise linear, many calculations can be performed exactly, unlike for the Mathieu equation. When ${\displaystyle a=b=1}$, the Floquet exponents are roots of the quadratic equation

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