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I solved the homogeneous differential equation of a damped oscillator ((D@@2)(x))(t)+d*(D(x))(t)+k^2*x(t) = A*sin(omega*t) with maple, the output is:

x(t) = _C1*exp((-(1/2)*d+(1/2)*sqrt(d^2-4*k^2))*t)+_C2*exp((-(1/2)*d-(1/2)*sqrt(d^2-4*k^2))*t)

Now, as there is damping, the limit for t->infinity shoud be 0. I substituted:

hommod := subs(d^2-4*k^2 = Delta, rhs(l_hom))

 Then, I tried the limit(hommod, t = infinity) command for the three cases

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