Functional equations satisfied by families of linear bounded operators related to Cauchy problems imply that the trajectories of the families exhibit surprising regularity properties. It turns out that these trajectories tend to be the more regular the higher the order of the Cauchy problem is. The talk is devoted to differences between trajectories of semigroups of operators (related to the first order Cauchy problem) and those of cosine operator families (related to the second order Cauchy problem). In particular we show that in contrast to the rich theory of singular perturbations of semigroups, there are no objects to study in the theory of singular perturbations of cosine families. Similarly, while there is plenty of examples of asymptotically stable semigroups (i.e., semigroups {S(t),t≥0} for which the limit lim_t→∞S(t) exists in the strong topology), there is only one cosine family with this property, and this is the trivial family with all operators being the identity operator.