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## Homework Statement

A Force

*F(t) = F*, where both

_{0}(1 - e^{-at})*F*and

_{0}*a*are constants, acts over a damped oscillator. In

*t = 0*, the oscillator is in it's equilibrium position. The mass of the oscillator is

*m*, the spring constant is

*k = 2ma*and the damping constant is

^{2}*b = 2ma*.

Find

*x(t)*

## Homework Equations

Well, the differential equation is: d

^{2}x / dt

^{2}+ 2adx/dt + 2a

^{2}x = F

_{0}/m - F

_{0}/m * e

^{-at}

Also,

*x(0) = 0*;

*x'(0) = 0*.

## The Attempt at a Solution

First, I tried to find the solution to the homogeneous equation associated.

So I tried a solution of the kind

*x = e*, and the I found: p

^{pt}^{2}+ 2ap + 2 = 0

p = (-2a [tex]\pm[/tex] [tex]\sqrt{-4a^2}[/tex]) / 2

p = -a [tex]\pm[/tex] ia

So, it's an under-dampeing case, wich the general solution is A*e

^{-at}*cos(at + [tex]\theta)[/tex].

Then I have to find the particular solutions, where x

_{p1}= F

_{0}/m and x

_{p2}= -F

_{0}/m * e

^{-at}.

x

_{p1}is rather easy. C*x

_{p1}= F

_{0}/m ; 2a

^{2}*x

_{p1}= F

_{0}/m ; x

_{p1}= F

_{0}/2ma

^{2}

Now I'm

*stuck*at the other one. I tried solutions like x = C*e

^{pt}and x = C*t

^{2}*e

^{pt}(C = constant) and haven't got it right.

The answer is:

*x(t) = F*

_{0}/ma^{2}* [[tex]\sqrt{2}[/tex]*e^{-at}*cos(at + P/4) + 1 - 2e^{-at}]How do I find the other particular solution? What am I missing?

(By the way, should this be here or in the Calculus sub-forum?)

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