The determination of a particular solution for the systems of linear differential equations with constant coefficients that have on the column of free terms functions such as ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx, or/and eb·x·Q(x) sinnx, n ∈ N, is based on the expansion of cosn x and sinn x, and, on the other hand, on how a particular solution for a column of free terms with functions such as ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx, n ∈ N, looks like. We can also write the way a particular solution looks like when we have a combination of two or more functions on the form ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx, n ∈ N in the column of free terms.