We present a complete description of the form of transcendental meromorphic solutions of the second order differential equation † \(w^{\prime\prime}-w^{\prime2}+aw^{\prime} w+bw^2=\alpha+\beta w^\prime+\gamma\) 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 \(\varsigma (w)\le n\) 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 \(\gamma \not \equiv 0\) and w has infinite order, then the hyper-order \(\varsigma (w)\) is a positive integer.
