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On rectifiable oscillation of Euler type second order linear differential equations. (English) Zbl 1182.34049

Summary: We study the oscillatory behavior of solutions of the second order linear differential equation of Euler type:
\[ y'' + \lambda x^{-\alpha} y = 0, \;x \in (0, 1],\tag{E} \]
where \(\lambda > 0\) and \(\alpha> 2\).
Theorem (a) For \(2 \leq \alpha < 4\), all solution curves of \((E)\) have finite arc length; (b) For \(\alpha \geq 4\), all solution curves of \((E)\) have infinite arc length. This answers an open problem posed by M. Pasic.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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