# Damped Driven Harmonic Oscillator

## 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.

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

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

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

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

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

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

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