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**Apprentice123** The differential equation system is

$\displaystyle m \frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx(t) = F_o cos(wt)$

To solve this differential equation requires a particular solution xp(t) and two fundamental solutions of the homogeneous equation corresponding X1(t) and X2(t)

$\displaystyle x(t) = xp(t) + C1x1(t) + C2x2(t)$

I find X1(t) and X2(t) for the following cases

Overdamped:

$\displaystyle x(t) = C1e^{r1t} + C2e^{r2t}$

$\displaystyle r1 = \frac{ - \gamma + \sqrt{( \gamma )^2 -4mk}}{2m}$

$\displaystyle r2 = \frac{ - \gamma - \sqrt{( \gamma )^2 -4mk}}{2m}$

Underdamped

$\displaystyle x(t) = e^{-(\frac{c}{2m})t} (C1cos(rt) + C2sin(rt))$

r -> imaginary part

$\displaystyle r = \frac{ \sqrt{( \gamma )^2 -4mk}}{2m}$

How do I find the particular solution xp(t) ?