Mathieu Equation

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Kushal Shah - 07 Oct 2008 12:26 GMT

Hello,

Mathieu equation is a well studied equation:

y''+(p-2qcos(2x))y=0

But can someone please suggest methods to find the particular solution
to this equation:

y''+(p-2qcos(2x))y=b

where 'b', 'p' and 'q' are non-zero constants. I guess, the general
solution should be a sum of the particular solution of the above
equation and solution of the Mathieu equation.

Thanks,
Kushal.

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G A Edgar - 07 Oct 2008 18:21 GMT

> Hello,
>
[quoted text clipped - 13 lines]
> Thanks,
> Kushal.

What happens if you apply the method of variation of parameters?

Maple yields the following particular solution, which looks like it was
obtained that way:

y(x) = b*((Int(MathieuS(p, q, x)/(-MathieuC(p, q, x)*MathieuSPrime(p, q, x)
+MathieuS(p, q, x)*MathieuCPrime(p, q, x)), x))*MathieuC(p, q, x)
-(Int(MathieuC(p, q, x)/(-MathieuC(p, q, x)*MathieuSPrime(p, q, x)
+MathieuS(p, q, x)*MathieuCPrime(p, q, x)), x))*MathieuS(p, q, x))

[variation of parameters:
http://www.math.ohio-state.edu/~edgar/movie/DayTheEarth.html
]

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G. A. Edgar http://www.math.ohio-state.edu/~edgar/

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Math Forum / Mathematics / Research / October 2008

Mathieu Equation