The n-dimensional differential system z'= (R + A(t))z is considered, where R is a constant n x n complex matrix and A(t) is an n x n matrix whose entries a(t) are complex valued functions which are representable as absolutely convergent Laplace-Stieltjes transforms, j e-St cda(s), for t > 0. The determining functions, a(s), are C valued, locally of bounded variation on LO, oo), continuous from the right, and a(+ 0) = a(0) =0. Sufficient conditions on the determining functions are found which assure the existence of solutions of certain specified forms involving absolutely convergent Laplace-Stieltjes transforms for t > 0 and which behave asymptotically like certain solutions of the nonperturbed equation z2 = Rz as t -oo. Analogous results are proved for the nth order equation n . (D r)e(i) + n a (t)Diz 0, where r. e C and the a .(t) are like a(t) above for t > 0. 0. Introduction. Consider the n-dimensional, first order, linear, ordinary differential system (0.1) y' = (R + A(t))y, where R is an n x n constant complex matrix and A(t) is an n x n matrix whose entries are complex valued functions representable as absolutely convergent Laplace-Stieltjes (or Laplace) transforms, A(t)=f0 e-stda(s) for t>0. Received by the editors May 21, 1973 and, in revised form, June 29, 1973. AMS (MOS) subject classifications (1970). Primary 34A30, 34D05; Secondary 34A10, 34A25, 34E10.