  Let $P(z)$ be a polynomial of degree $n\geq 1$ and $S(z)$ be a nonzero
rational function. It is shown that if $f(z)$ is a meromorphic solution of the
first order differential equation $f'(z)=S(z)e^{P(z)}f(z)+1$, then there is a
curve $\Gamma: x\to x+iy(x)$, where $x_0\leq x<\infty$ and
$\pi<nx^{n-1}y<3\pi/2$ such that for all $z\in \Gamma$,
\begin{equation}\tag{\dag} |f(z)|>
\exp\left(\frac{1}{8}e^{\frac{1}{8}x^{n}}\right). \end{equation} The estimate
in $(\dag)$ together with the Wiman--Valiron theory implies that the
hyper-order $\varsigma(f)$ of $f(z)$ is equal to $n$, which allows to specify
the class $N$ in complex dynamics. Moreover, this provides partial answers to
Br\"{u}ck's conjecture in uniqueness theory and also a problem on a second
order algebraic differential equation of Hayman.
