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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 '' +λx -α y=0,x(0,1],(E)

where λ>0 and α>2.

Theorem (a) For 2α<4, all solution curves of (E) have finite arc length; (b) For α4, all solution curves of (E) have infinite arc length. This answers an open problem posed by M. Pasic.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general