Nonlinear differential equations with transcendental meromorphic solutions (Q2726652)

Consider the nonlinear differential equations NEWLINE\[NEWLINE \begin{aligned} &(f')^n=q(z)P(f)P_0(f')(f-z)^m, \tag{1}\\ &(f')^n=q(z)e^{P_1(z)}P(f)(f-z)^m, \tag{2} \end{aligned} NEWLINE\]NEWLINE with \(m,n\in \mathbb{N}\), \(q(z)\) is a rational function, and \(P(z),\) \(P_0(z)\) and \(P_1(z)\) are polynomials satisfying \(P_0(0)\not=0.\) Here, the authors give complete classifications of these equations under the supposition that each equation admits a transcendental meromorphic solution. For example, if equation (1) has such a property, then it must be one of the equations below: NEWLINE\[NEWLINEf'=q(z)(f-z), \quad f'=q(z)(f-z)^2, \quad f'=q(z)(f-\tau)(f-z),NEWLINE\]NEWLINE NEWLINE\[NEWLINE(f')^2=q(z)(f-\tau)(f-z)^2,\quad (f')^2=q(z)(f-\tau)(f-\delta)(f-z)^2,NEWLINE\]NEWLINE with \(\tau,\delta\in\mathbb{C},\) \(\tau\not=\delta.\) In the proofs, some lemmas concerning the Nevanlinna theory are employed.