On the oscillation of certain second-order linear differential equations

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 $k\geq 1$ 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 $n\geq 2$ is an integer, $P(z,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha=[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a non-trivial solution such that $\lambda(f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not=0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max\{\lambda(f_1),\lambda(f_2)\}<\infty$. The method for the proof also leads to a positive answer to Ishizaki's question in the case $b_3=0$. In particular, we precisely characterize all solutions such that $\lambda(f)<\infty$ when $l=2$ and $l=4$.Comment: 25page