In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and L-2[0, infinity)-solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) . An example is given to illustrate the obtained results. (C) 2015 Production and hosting by Elsevier B.V. on behalf of Cairo University.