  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 nontrivial 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
$\max\{\lambda(f_1),\lambda(f_2)\}<\infty$. In particular, we precisely
characterize all solutions such that $\lambda(f)<\infty$ when $l=2$ and $l=4$.
