On transcendental meromorphic solutions of Hayman's equation
Yueyang Zhang
TL;DR
This work analyzes the second-order ODE $w''w - w'^2 + a w'w + b w^2 = \alpha w + \beta w' + \gamma$ with rational coefficient functions, aiming to classify all transcendental meromorphic solutions. By integrating Nevanlinna theory with Wiman–Valiron methods, the authors derive a universal growth bound: the hyper-order satisfies $\varsigma(w) \le n$ for some integer $n \ge 0$, and finite-order solutions have $2\sigma(w) \in \mathbb{N}$, with stronger integer-valued hyper-orders in certain cases; Bank's conjecture is shown to hold for this equation. They then present a complete form description (Theorem maintheorem0) of all transcendental meromorphic solutions, splitting into multiple explicit families depending on whether $\alpha$, $\beta$, $\gamma$ vanish and on the structure of auxiliary functions, including reductions to first-order linear ODEs and cosh/exponential-type solutions; the analysis accounts for algebroid factors and conditions under which exponential factors are meromorphic or rational. The results generalize prior work for special coefficient choices (e.g., $a=b=0$) to the case of general rational coefficients and provide concrete, solvable forms that illuminate the differential-algebraic structure of Hayman's equation. As a corollary, Bank's growth conjecture is validated for this broader class of second-order ODEs, highlighting the interplay between coefficient rationality, growth, and explicit solution representations.
Abstract
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 $a$, $b$, $α$, $β$ and $γ$ are all rational functions. Together with the Wiman--Valiron theory, we then show that any transcendental meromorphic solution $w$ of equation $(†)$ has hyper-order $ς(w)\leq n$ for some integer $n\geq 0$. Moreover, if $w$ has finite order $σ(w)$, then $2σ(w)$ is a positive integer; if $β\equivγ\equiv0$ and $w$ has infinite order or if $γ\not\equiv0$ and $w$ has infinite order, then the hyper-order $ς(w)$ is a positive integer.