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We organize this work as follows: In first part we study the relative asymptotic equivalence, with weight t^µ, where µ is a non negative integer number, of systems: (a) y = A(t)y (b) x = A(t)x + f(t, x), namely, we prove that under certain assumptions, the followings results hold: 1) For every solution y(t)≠ O of (a) there is at least one solution x(t) of (b) satisfying: (c) t^µ ΙΙ x (t) - y(t) ΙΙ / ΙΙ y (t) → 0, as t → 0 ∞ For every solution x(t) of (b), x(t); ≠ O for all sufficiently large t, there corresponds at least one solution y(t) of (a) satisfying (c). We also give information about the number of parameters of the family of solutions x(t) of (b) satisfying 1). A similar result follows with respect condition. 2) In second part we apply the above mentioned results to a class of matrices A(t) which contains the periodic and constant matrices as particular cases.