Oscillation with Green's Function

[Menteith]

Homework Statement

A force F_ext(t) = F₀[ 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 q² 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.

Homework Equations

Green's Function:
x(t) = ∫_-∞^tF(t')G(t,t')dt'
where:
G(t,t') = (1/(m*ω₁))*e^-β(t-t')*sinω₁(t-t') for t≥t'
= 0 for t<t'

The Attempt at a Solution

I have solved using green's function to obtain this mess:
(F₀/m)*(e^-βt/(β²+ω₁²))*((β/ω₁)*sinω₁t - cosω₁t) - (F₀/m)/((α-β)²+ω₁²)*[e^-αt-e^-βt*(cosω₁t-((α-β)/ω₁)sinω₁t)]

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

 

[Couchyam]

The final position can be found by balancing the limiting force with Hook's law.

 

[Charles Link]

I would suggest you work the problem with a simple differential equation. The Green function approach for this one is rather difficult. If your instructor specified he requires a Green's function type solution, then it is the route you need to go, but for this one, the Green's function looks like a difficult approach. editing... I'm looking at the date on the OP. This one appears to be a couple of years old.