On the oscillatory integration of some ordinary differential equations

TL;DR

This paper establishes conditions under which certain nonlinear ordinary differential equations admit solutions with specific asymptotic behavior and oscillatory properties, extending understanding of solution structures in differential equations.

Contribution

It provides new criteria for solutions with prescribed asymptotes and oscillatory pseudo-Wronskian in a class of nonlinear ODEs, including linear cases.

Findings

Solutions with prescribed oblique asymptote exist under certain conditions.

The solutions exhibit oscillatory pseudo-Wronskian behavior.

The results encompass linear and nonlinear differential equations.

Abstract

Conditions are given for a class of nonlinear ordinary differential equations x''(t)+a(t)w(x)=0, t>=1, which includes the linear equation to possess solutions x(t) with prescribed oblique asymptote that have an oscillatory pseudo-wronskian x'(t)-x(t)/t.