Geodesic
On the oscillatory integration of some ordinary differential equations
Archivum mathematicum, Tome 44 (2008) no. 1, pp. 23-36.

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Conditions are given for a class of nonlinear ordinary differential equations $x^{\prime \prime }+a(t)w(x)=0$, $t\ge t_0\ge 1$, which includes the linear equation to possess solutions $x(t)$ with prescribed oblique asymptote that have an oscillatory pseudo-wronskian $x^{\prime }(t)-\frac{x(t)}{t}$.
Classification : 34C10, 34D05, 34E05, 34K25
Keywords: ordinary differential equation; asymptotic integration; prescribed asymptote; non-oscillation of solutions 
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Mustafa, Octavian G. On the oscillatory integration of some ordinary differential equations. Archivum mathematicum, Tome 44 (2008) no. 1, pp. 23-36. http://geodesic.mathdoc.fr/item/ARM_2008__44_1_a3/