Example 5.13  p.77

gs = 
        1/(s^2 + a2 s + a1) /. {a2 -> 2, 
            a1 -> 5};
y0s = v0  /. v0 ->  -3;

para = U ->  UnitStep;
qt = U[t] - U[t - 1] /. para
qs = 5  Simplify[ LaplaceTransform[qt, t, s]]

yt2 = 
        Re[InverseLaplaceTransform[gs qs, s, 
            t]];
yt = 
        yt2 + InverseLaplaceTransform[gs y0s, s, 
            t];
Plot[{qt, yt2}, {t, 0, 5}]

Plot[{qt, yt2, yt}, {t, 0, 5}]

example 5.14   Fig. 5.21  p.78

q[t_] := Sin[w t]  U[t] /. para;
qt = q[t] + q[t - Pi/w] /. w -> Pi
qs = 5  Simplify[ LaplaceTransform[qt, t, s]]

yt2 = 
        Re[InverseLaplaceTransform[gs qs, s, 
            t]];
yt = 
        yt2 + InverseLaplaceTransform[gs y0s, s, 
            t];
Plot[{qt, yt2, yt}, {t, 0, 5}, 
      PlotRange ->  {-0.8, 1}]

example 5.14   Fig. 5.21  p.78

q[t_] :=  t  U[t] /. para;
qt = 
      q[t] - 2 q[t - a] + q[t - 2  a] /. 
        a -> 1
qs = 5  Simplify[ LaplaceTransform[qt, t, s]]

yt2 = 
        Re[InverseLaplaceTransform[gs qs, s, 
            t]];
yt = 
        yt2 + InverseLaplaceTransform[gs y0s, s, 
            t];
Plot[{qt, yt2, yt}, {t, 0, 5}, 
      PlotRange -> {-1, 1}]