On transcendental meromorphic solutions of Hayman's equation

We present a complete description of the form of transcendental meromorphic solutions of the second order differential equation \begin{equation}\tag{\dag} w''w-w'^2+a w'w+b w^2=\alpha w+\beta w'+\gamma, \end{equation} where $a$, $b$, $\alpha$, $\beta$ and $\gamma$ are all rational functions. Together with the Wiman--Valiron theory, we then show that any transcendental meromorphic solution $w$ of equation $(\dag)$ has hyper-order $\varsigma(w)\leq n$ for some integer $n\geq 0$. Moreover, if $w$ has finite order $\sigma(w)$, then $2\sigma(w)$ is a positive integer; if $\beta\equiv\gamma\equiv0$ and $w$ has infinite order or if $\gamma\not\equiv0$ and $w$ has infinite order, then the hyper-order $\varsigma(w)$ is a positive integer.