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**1. Homework Statement**

A force F

_{ext}(t) = F

_{0}[ 1−e

^{(−αt)}] acts, for time t > 0, on an oscillator which is at rest at x=0 at time 0.

The mass is m; the spring constant is k; and the damping force is −b x′. The parameters satisfy these relations:

b = m q , k = 4 m q

^{2}where q is a constant with units of inverse time.

Find the motion. Determine x(t); and hand in a qualitatively correct graph of x(t).

(B) Determine the final position.

**2. Homework Equations**

Green's Function:

x(t) = ∫

_{-∞}

^{t}F(t')G(t,t')dt'

where:

G(t,t') = (1/(m*ω

_{1}))*e

^{-β(t-t')}*sinω

_{1}(t-t') for t≥t'

= 0 for t<t'

**3. The Attempt at a Solution**

I have solved using green's function to obtain this mess:

(F

_{0}/m)*(e

^{-βt}/(β

^{2}+ω

_{1}

^{2}))*((β/ω

_{1})*sinω

_{1}t - cosω

_{1}t) - (F

_{0}/m)/((α-β)

^{2}+ω

_{1}

^{2})*[e

^{-αt}-e

^{-βt}*(cosω

_{1}t-((α-β)/ω

_{1})sinω

_{1}t)]

From here, however, I am unsure of how to find the final position without the final time.

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