Asymptotic behavior of solutions of a nonlinear differential equation of the first order

Abstract

The asymptotic behavior at infinity of solutions of the equation u′ = P(u, t) Q(u, t) is studied. P, Q are polynomials in u whose coefficients are functions of t, and belong to the Hardy class H (i.e., to the set of all real-valued functions defined by finite many ordinary algebraic, exp, and log operations.) It is proved that, for any continuously differentiable solution u(t), there exists one or the other of the asymptotic formulae u(t) ~ h(t), In u(t) ~ h(t), within the class H, i.e., h(t) ε{lunate} H. As the main tool for the proof it is first shown that any (real) solution y(t) of an algebraic equation whose coefficients are elements of H behaves at infinity again as an element of H. © 1972.