General Solution to Mathieu's Equation

Theorem

The general solution to Mathieu's equation:

$\dfrac {\d^2 y} {\d x^2} + \paren {a + b \cos 2 x} y = 0$

is:

$A e^{r x} \map \phi x + B e^{-r x} \map \phi {-x}$

where $r$ is a constant and $\phi$ a periodic function of period $2 \pi$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Mathieu's equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Mathieu's equation