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Conditions are given for a class of nonlinear ordinary differential equations $x^{\text{'}\text{'}}+a\left(t\right)w\left(x\right)=0$, $t\ge t_0\ge 1$, which includes the linear equation to possess solutions $x\left(t\right)$ with prescribed oblique asymptote that have an oscillatory pseudo-wronskian $x^{\text{'}}\left(t\right)-\frac{x\left(t\right)}{t}$.