On a question of Gundersen-Yang concerning entire solutions of binomial differential equations

TL;DR

This paper classifies entire solutions to specific binomial differential equations involving polynomial coefficients using Nevanlinna theory, partially answering a question posed by Gundersen and Yang.

Contribution

It provides explicit forms of entire solutions for certain binomial differential equations, advancing understanding of their solution structure.

Findings

Explicit solutions for the differential equations are obtained.

Partial answers to Gundersen and Yang's question are provided.

Examples illustrate the theoretical results.

Abstract

We study the question posed by G. Gundersen and C. C. Yang, in which the following two types of binomial differential equations are investigated, $$ a(z)f'f''-b(z)(f)^{2}=c(z)e^{2d(z)},~~a(z)ff'-b(z)(f'')^{2}=c(z)e^{2d(z)}, $$ where $a(z)$, $b(z)$ and $c(z)$ are polynomials such that $a(z)b(z)c(z)\not\equiv 0$, $d(z)$ is non-constant polynomial. The explicit forms of entire solutions of the above binomial differential equations are obtained by using the Nevanlinna theory, which gives partial solutions to the question of G. Gundersen and C. C. Yang. In addition, some examples are given to illustrate these results.