Newton's second law leads to the differential equation:
m d2x/dt2 + k x = F0 e-αt
The solution to the homogeneous equation is the familiar
x = A cos(ω t) + B sin(ω t) where A and B are constants to be determined by initial conditions.
( with ω = sqrt(k/m) )
To get a full solution, a particular solution to the inhomogeneous equation is needed. Any form that
works will do. The full solution is then the sum of this particular solution and the
A cos + B sin form above.
To get such a particular solution, make an ansatz : x = Q e-αt
On plugging this into the the full differential equation, it is seen that it works
if Q = F0 /( m α2 + k) Thus the particular solution is
x = [F0 /(m α2 + k) ] e-αt
