On meromorphic solutions of the Riccati differential equations
Volume 99 / 2010
Annales Polonici Mathematici 99 (2010), 247-262 MSC: Primary 34A34; Secondary 34M05, 30D35. DOI: 10.4064/ap99-3-3
We investigate the growth and Borel exceptional values of meromorphic solutions of the Riccati differential equation $$ w'=a(z)+b(z)w+w^2, $$ where $a(z)$ and $b(z)$ are meromorphic functions. In particular, we correct a result of E. Hille [Ordinary Differential Equations in the Complex Domain, 1976] and get a precise estimate on the growth order of the transcendental meromorphic solution $w(z)$; and if at least one of $a(z)$ and $b(z)$ is non-constant, then we show that $w(z)$ has at most one Borel exceptional value. Furthermore, we construct numerous examples to illustrate our results.