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We deal with the basic ordinary differential systems y = (1) y = A(t)y (2) x = A(t)x + f(t, x) where x, y and f(t, x) are n-vectors, A(t) is an n x n matrix and t ranges on (t₀ , ∞), t₀ ≥ 0. If µ ≥ 0 is an integer and ρ ≥ 0 is a real, we give conditions on f(t, x) to obtain a positive answer to the following problems: (I) If y(t) ≠ 0 is a solution of (1), find a family of solutions x(t) of (2) satisfying lim_t→∞ t^µ e^ρt x(t) - y(t) / y(t) = 0 (II) Of x(t) is a solution of (2) with x(t) ≠ 0 for t ≥ t₀, find a family of solutions y(t) of (1) satisfying lim_{t→∞ t^µ e ^ρt x(t) - y(t) / x(t) = 0. We also give information, in each case, about the number of parameters depending the family of solutions obtained. ne plan this work as follows: In the first part, we study the A(t) -admissibility of a pair (B,D) of Banach spaces and give a positive answer to the above roblems. We also derive a result on relative asymptotic equivalence, with weight t^µ e^ρt, between two perturbed systems of (1). In the second part, we restrict the conditions On fít,x) and make the above study, with additional information about uniqueness of solution. This enables us to obtain certain topological property on the initial values of the families of solutions found in problems (I) and (IT).}