Damped Driven Harmonic Oscillator

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Homework Statement



I'm trying to follow through a derivation involving the equation of motion for the displacement x(t) of a damped driven harmonic oscillator.

[tex]m\frac{d^{2}x}{dt^{2}}+\gamma x + \beta \frac{dx}{dt}=F_{0}cos(\omega t)[/tex]

Where
[tex]cos(\omega t) = \frac{1}{2}\left( e^{i \omega t} + e^{-i \omega t} \right)[/tex]

Look for a solution of the form [tex]x(t) = X^{+}(\omega)e^{i \omega t} + X^{-}(\omega)e^{-i \omega t}[/tex]

Solve for [tex]X^{-}(\omega)[/tex] and note that [tex]X^{+}(\omega)=(X^{-}(\omega))^{*}[/tex] because x(t) is a real quantity.

Substitute [tex]x(t) = X^{-}(\omega)e^{-i \omega t}[/tex] into the equation of motion:

[tex]mX^{-}(\omega)(-\omega^{2})e^{-i \omega t} + \gamma X^{-}(\omega)e^{-i \omega t} - i\omega\beta X^{-}(\omega)e^{-i \omega t} = \frac{F_{0}}{2}e^{-i \omega t}[/tex]


I understand the crux of the derivation however I'm not sure where the [tex]\frac{1}{2} e^{i \omega t}[/tex] has gone from the cosine term. The negative exponent has cancelled, but the positive just seems to have disappeared. Could anyone help!? Thanks!
 

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