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This paper aims to study the relative equivalence of the solutions of the following dynamic equations y Δ (t) = A (t) y (t) and x Δ (t) = A (t) x (t) + f (t , x (t)) in the sense that if y (t) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions x (t) for the perturbed system such that ∥ y (t) − x (t) ∥ = o (∥ y (t) ∥) , as t → ∞ , and conversely, given a solution x (t) of the perturbed system, we give sufficient conditions for the existence of a family of solutions y (t) for the unperturbed system, and such that ∥ y (t) − x (t) ∥ = o (∥ x (t) ∥) , as t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales. [ABSTRACT FROM AUTHOR]