The two-dimensional systems of first order nonlinear differential equations
(S1) x? = p(t)y?, y? = q(t)x? and (S2) x? + p(t)y? = 0, y? + q(t)x? = 0 are
analyzed using the theory of rapid variation. This approach allows us to
prove that all strongly increasing solutions of system (S1) (and,
respectively, all strongly decreasing solutions of system (S2) ) are rapidly
varying functions under the assumption that p and q are rapidly varying.
Also, the asymptotic equivalence relations for these solutions are given.