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Our derivation below can be found in greater detail and better form
in many references [3,4,5], and our
derivation follows the spirit of these. An equation such as
Mathieu's equation,
![$\displaystyle \ddot{\eta}+(a-2q\cos(2\tau))\eta=0$](img18.png) |
(2.1) |
is of a class of differential equations of the type [7],
![$\displaystyle L[y]=y''+p(t)y' + q(t)y = 0$](img19.png) |
(2.2) |
Any two fundamental solutions to this equation,
, will satisfy
the set of boundary value equations,
We thus require that the determinant of Y (called the Wronskian) is not equal to zero,
![$\displaystyle W(Y)=det(Y)=\left\vert\begin{array}{cc}y_1(t_0)&y_2(t_0)\ y'_1(t_0)&y'_2(t_0)\end{array}\right\vert \neq 0$](img22.png) |
(2.3) |
The set of even/odd solutions:
Are thus fundamental sets of solutions. We may follow Floquet's theorem [3], which
tells us that Mathieu's equation has at least one solution
Floquet's Theorem |
(2.4) |
The proof of this is outlined as follows.
Since Mathieu's equations will have an even (
) and odd (
)
solution pair, these two functions may define any other solution, e.g. consider
![$\displaystyle w_1(\eta+\pi)=w_1(\pi)w_1(\eta)+w'_1(\pi)w_2(\eta), $](img30.png)
as well,
Let
According to Floquet's theorem, we thus require,
an eigenvalue equation which can be satisfied with the proper value of
.
We consider further that Mathieu's equation has a solution of the form
Next: Hill's Method solution
Up: Mathieu's Equation, solution, and
Previous: Mathieu's Equation, solution, and
tim jones
2008-07-07