On entire solutions of a certain type of nonlinear differential equations (Q2822571)

Let \(\mathbb{C}\) denote the complex plane and \(\mathbb R\) denote the ring of entire in \(\mathbb C\) functions. The paper deals with the differential equation of the form NEWLINE\[NEWLINEE(y)=p_1e^\alpha_1z+p_2e^\alpha_2z,NEWLINE\]NEWLINE where \(E\in\mathbb C[z]\{y\}\), \(p_1,p_2\in\mathbb C[z]\backslash 0\), \(\alpha_1,\alpha_2\in\mathbb C\backslash 0\). The authors address the following question: When has this equation no solution in \(\mathbb R\backslash\mathbb C[z]\) (see [\textit{P. Li} and \textit{C.-C. Yang}, J. Math. Anal. Appl. 320, No. 2, 827--835 (2006; Zbl 1100.34066)])? The authors show that if the differential polynomial \(E\) can be presented in the form \(E(y)=y'y^n+P(y)\), \(P\in\mathbb C[z]\{y\},\deg P<n\), the answer will be positive in the following two cases:NEWLINENEWLINE(i) \(n\geq 4\), \(\deg P\leq n-3\), \(\frac{\alpha_1}{\alpha_2}\neq(\frac{k}{n+1})^{\pm 1}\), \(1\leq k\leq\deg P\) and \(\frac{\alpha_1}{\alpha _2}\neq 1\);NEWLINENEWLINE(ii) \(n\geq 3\), \(\deg P\leq n-2\), \(\frac{\alpha_1}{\alpha_2}=-1\) and \(p_1,p_2\in\mathbb C\backslash 0\).