Oscillation properties of 𝑦ⁿ+𝑝𝑦=0

Jones, Gary D.

doi:10.1090/S0002-9939-1980-0550504-3

Oscillation properties of $y^{n}+py=0$
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by Gary D. Jones

Proc. Amer. Math. Soc. 78 (1980), 239-244

DOI: https://doi.org/10.1090/S0002-9939-1980-0550504-3

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Abstract:

The purpose of this paper is to give necessary and sufficient conditions for $(k,n - k)$ disconjugacy of ${y^n} + py = 0$. The results are then applied to give counterexamples to the following theorems of Nehari. If n is even and p is positive either all solutions of ${y^n} + py = 0$ oscillate or none do. If n is even and p is negative and ${y^n} + py = 0$ has an oscillatory solution then all positive nonoscillatory solutions are either strongly increasing or strongly decreasing.

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