On the oscillation of certain second-order linear differential equations

Abstract

This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\ge 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura--Clunie type differential equation $f^n+P(z,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\ge 2$ is an integer, $P(z,f)$ is a differential polynomial in $f$ of degree $\le n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f^{\prime \prime }-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\ge 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\ne 0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l>s\ge 1$, we prove that $l=2$ if the equation $f^{\prime \prime }-(e^{lz}+b_2{e}^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

arXiv comment25pages

Cited by in corpus (1)

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