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† w’‘w-w’^2+a w’w+b w^2=αw+βw’+γ, \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 $(\text{†})$ has hyper-order $\varsigma (w)\le n$ for some integer $n\ge 0$. Moreover, if $w$ has finite order $\sigma (w)$, then $2\sigma (w)$ is a positive integer; if $\beta \equiv \gamma \equiv 0$ and $w$ has infinite order or if $\gamma \equiv ̸0$ and $w$ has infinite order, then the hyper-order $\varsigma (w)$ is a positive integer.