Particular differnetial solution

[Phymath]

solve for the particular solution of the damped harmonic oscillator driven by the damped harmonic force
$$F(t) = F_0e^{-\alpha t)cos(\omega t)$$

(Hint: $$e^{-\alpha t} cos(\omega t) = Re[e^{-\alpha t + i \omega t}] = Re[e^{B t}]$$ where $$B = -\alpha + i \omega$$. Find the solution in the form $$x(t) = De^{B t - i \phi}$$, i don't have much diff e q the only thing i think of doing is the following..

$$x'' + 2 \gamma x' + \omega^2 x = 0$$
$$c^2 + 2 \gamma c + \omega^2 = 0$$
$$x(t) = C_1 e^{-(\sqrt{\gamma^2 - \omega^2}+\gamma)t} + C_2 e^{(\sqrt{\gamma^2 -\omega^2}-\gamma) t}$$
no idea where to go from here... any help would be awesome

 

[CarlB]

I forget what the damped harmonic oscillator DE is but it looks like you solved the homogenous part of the equation. Now all you need is a single solution to the inhomogenous equation.

To do this, follow the hint they've given you, take a wild chance and just substitute $Re[e^{Bt}]$ for x. You won't get the right answer (that is, get F(t) on the RHS of your DE), but you should get something that is proportional and that will give you enough hint to get you through.

Good luck. It's a lot of work. Feel proud. Time for me to get some sleep.

Carl