A note on oscillatory integration (Q928088)

A Wronskian-like quantity \[ W(x,t)=\frac{1}{t}\left| \begin{matrix} t & x(t)\\ 1 & x^{\prime}(t) \end{matrix} \right| \] plays an important role in the study of solutions of several classes of differential equations with prescribed asymptotic behavior. The purpose of this interesting note is to derive sufficient conditions in terms of the coefficient \(a(t)\) that guarantee existence of a solution \(x(t)\) of the simple second order linear differential equation \[ x^{\prime\prime}+a(t)x=0,\qquad t\geq t_{0}\geq1, \] with the asymptotic representation \[ x(t)=c_{1}t+c_{2}+o(1) \text{ as } t\rightarrow\infty \] such that \(W(x,t)\) is oscillatory (in the usual sense). The main result can be applied for the study of bounded and periodic solutions to certain classes of elliptic differential equations.