Asymptotic relationships between two higher order ordinary differential equations

Some asymptotic relationships between the two ordinary differential equations (1) x(n)+p1(t)x(n−1)+…+pn(t)x=0, (2) y(n)+p1(t)y(n−1)+…+pn(t)y=f(t,y), are studied. Conditions are given that lead to an asymptotic equivalence between certain of the solutions of (1) and certain of the solutions of (2). The case where the perturbation f(t,y) depends on a functional argument is also discussed