  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
$a,b,\alpha,\beta,\gamma$ are all rational functions. Together with the
Wiman--Valiron theorem, our results yield that any transcendental meromorphic
solution $w$ of $(\dag)$ has hyper-order $\varsigma(w)\leq n$ for some integer
$n\geq 0$. Moreover, if $w$ has finite order $\sigma(w)=\sigma$, then $\sigma$
is in the set $\{n/2: n=1,2,\cdots\}$ and, if $w$ has infinite order and
$\gamma\not\equiv0$, then the hyper-order $\varsigma$ of $w$ is in the set
$\{n: n=1,2,\cdots\}$. Each order or hyper-order in these two sets is attained
for some coefficients $a,b,\alpha,\beta,\gamma$.
