Geodesic
An integral criterion for oscillation of linear differential equations of second order
Matematičeskie zametki, Tome 23 (1978) no. 2, pp. 249-252

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It is proved that if for some $n>2$ the function $x^{1-n}A_n(x)$, where $A_n(x)$ is the $n$-th primitive of $a(x)$, is not bounded above, then the equation $y''+a(x)y=0$ oscillates. 
@article{MZM_1978_23_2_a7,
     author = {I. V. Kamenev},
     title = {An integral criterion for oscillation of linear differential equations of second order},
     journal = {Matemati\v{c}eskie zametki},
     pages = {249--252},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_2_a7/}
}
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I. V. Kamenev. An integral criterion for oscillation of linear differential equations of second order. Matematičeskie zametki, Tome 23 (1978) no. 2, pp. 249-252. http://geodesic.mathdoc.fr/item/MZM_1978_23_2_a7/