  Let $p(z)$ be a nonconstant polynomial and $\beta(z)$ be a small entire
function of $e^{p(z)}$ in the sense of Nevanlinna. We first describe the growth
behavior of the entire function $H(z):=e^{p(z)}\int_0^{z}\beta(t)e^{-p(t)}dt$
on the complex plane $\mathbb{C}$. As an application, we solve entire solutions
of Tumura--Clunie type differential equation
$f(z)^n+P(z,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $b_1(z)$ and $b_2(z)$
are nonzero polynomials, $p_1(z)$ and $p_2(z)$ are two polynomials of the same
degree~$k\geq 1$ and $P(z,f)$ is a differential polynomial in $f$ of degree
$\leq n-1$ with meromorphic functions of order~$<k$ as coefficients. These
results allow us to determine all solutions with relatively few zeros of the
second-order differential equation
$f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}+b_3(z)]f=0$, where $b_3(z)$ is a
polynomial. We also prove a theorem on certain first-order linear differential
equation related to complex dynamics.
