FullSimplify[
      DSolve[{y''[t] + 2 y'[t] + 5  y[t] == 12 Exp[-t], 
          y[0] == 0, y'[0] == 0}, y[t], t]]

Plot[{Exp[-t] (-3 Cos[2 t] + 3), 
        6 (E^(-t)) Sin[t]^2}, {t, 0, 3  Pi}, 
      PlotRange -> {-2, 2}]

FullSimplify[
      DSolve[{y''[t] + 2 y'[t] + 5  y[t] ==  Cos[t], 
          y[0] == 0, y'[0] == 0}, y[t], t]]

Plot[{(1/10) (E^(-t)) (-2 Cos[2 t] - 
              3 Cos[t] Sin[
                  t] + (E^t) (2 Cos[t] + Sin[t]))}, {t, 
        0, 10  Pi}, PlotRange -> {- .5,  .5}]