Asymptotics of Nonoscillatory Solutions of Some Second‐Order Linear Differential Equations

In this paper we prove a theorem on the existence and asymptotic behaviour of nonoscillatory solutions of the equationx ″ + p ( t ) x = 0 , where p ( t ) = (−λ 2 + h ( t )) t −2α with λ > 0 and 0 < α < 1. The coefficient p need not be onesigned. Examples show that the same asymptotic formula can hold either when p ( t ) is eventually negative, or when it oscillates. Moreover, the result can be applied to cases where p is negative, but is such that the classical Liouville‐Green approximation formula cannot be used.