Asymptotic integration of nonoscillatory second-order differential equations (Q630247)

This paper first discusses a sufficient condition for the approximate solution of the second-order differential equation \[ u''-(1+\varphi(t))u=0 \] in which \(\varphi(t)\) is small for large values of \(t\) in some sense. The method is to reduce the above equation to an equivalent form similar to the Liouville-Green approximation using the Liouville transformation. Then, the author gives two asymptotic integrations of the above differential equation that are based on Levinson's theorem about L-diagonal systems and a reduction to the Riccati equation, respectively. Moreover, these asymptotic integrations are proved to be asymptotically equivalent in the sense that the estimates for their residual terms are identical on the common domain of their validity.