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 have the same degree $k\geq 1$ distinct leading coefficients $1$ $\alpha$ respectively, we solve entire solutions 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, $P(z,\,f)$ a polynomial in $f$ $\leq n-1$ with having growth. Second, study oscillation second-order $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ prove $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this admits nontrivial solution $\lambda (f)<\infty$ . partially answers question Ishizaki. Finally, $b_2\not =0$ $b_3$ constants $l$ $s$ relatively prime integers $l> s\geq $l=2$ $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ two linearly independent $f_1$ $f_2$ $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$ In particular, precisely characterize all when $l=4$