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Math Help - Annihilator Operators

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    Annihilator Operators

    If I have a linear, non-homogeneous differential equation with a function like e^{2x} on the right-side, one of the standard methods is to use an annihilater to transform it to a homogeneous equation.

    The operator (D-\alpha) annihilates functions of this form. However, it's also true that [D^2-2\alpha D+\alpha ^2+\beta ^2]^n annihilates fucntions of the form x^{n-1}e^{\alpha x}cos(\beta x). Using \alpha=2 and \beta =0 and n=1 gives the annihilater (D^2-4D+4)=(D-2)(D-2), so the corresponding auxiliary equation will contain 2 repeated roots for these factors. However, the annihilater (D-2) implies one root.

    I guess in general you can annihilate functions many ways, but using different annihilaters will give different auxiliary equations right?

    EDIT: Wait, \beta >0 is an assumption when deriving the operator, so what I'm doing doesn't make any sense. It also applies to the case that the function is a sine instead of a cosine, which would require a different value of \beta too. However, The differential operator I obtained still seems to work. I tested it.
    Last edited by adkinsjr; February 25th 2011 at 03:09 PM.
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