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

Jianren Long, Mengting Xia, Xuxu Xiang

TL;DR

The paper addresses the problem of characterizing all entire solutions to the binomial differential equations a(z)f'f''-b(z)f^2=c(z)e^{2d(z)} and a(z)ff'-b(z)(f'')^2=c(z)e^{2d(z)} with polynomial coefficients and non-constant d(z). Using Nevanlinna theory and differential-algebraic manipulations, it derives two main theorems that give exact forms for the solutions: either f(z)=p(z)e^{d(z)} with a precise polynomial relation, or f(x) is a finite exponential-sum under various parameter regimes; the results also handle degenerate cases with abc≡0 and remove prior restrictions such as f' having only simple zeros and λ(f)≤ρ(f) where applicable. The findings extend Gundersen–Yang’s partial results by providing complete classifications for (1.3) and (1.4) and include numerous explicit examples and the corresponding solution families. This advances the understanding of entire solutions to nonlinear binomial differential equations with polynomial data and highlights the role of exponential-polynomial structures in such problems.

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.

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

TL;DR

The paper addresses the problem of characterizing all entire solutions to the binomial differential equations a(z)f'f''-b(z)f^2=c(z)e^{2d(z)} and a(z)ff'-b(z)(f'')^2=c(z)e^{2d(z)} with polynomial coefficients and non-constant d(z). Using Nevanlinna theory and differential-algebraic manipulations, it derives two main theorems that give exact forms for the solutions: either f(z)=p(z)e^{d(z)} with a precise polynomial relation, or f(x) is a finite exponential-sum under various parameter regimes; the results also handle degenerate cases with abc≡0 and remove prior restrictions such as f' having only simple zeros and λ(f)≤ρ(f) where applicable. The findings extend Gundersen–Yang’s partial results by providing complete classifications for (1.3) and (1.4) and include numerous explicit examples and the corresponding solution families. This advances the understanding of entire solutions to nonlinear binomial differential equations with polynomial data and highlights the role of exponential-polynomial structures in such problems.

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, where , and are polynomials such that , 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.
Paper Structure (4 sections, 5 theorems, 107 equations)

This paper contains 4 sections, 5 theorems, 107 equations.

Table of Contents

  1. Introduction and main results
  2. Some Examples
  3. Proof of Theorem \ref{['the1']}
  4. Proof of Theorem \ref{['the2']}

Key Result

Theorem A

gy Let f be an entire solution of (eq1.1) Then we have one of the following situations: (1) $b(z)$ is non-constant and $f(z)=p(z)e^{c(z)}$, where p(z) is a polynomial. (2) $a(z) \equiv \tau$ for some constant $\tau\ne1$, $b(z)\equiv\mu$ for some constant $\mu\ne 0$, $c(z)=\lambda z$ for some constan where $\alpha$ is a constant satisfying $\alpha^2\lambda^2(1-\tau)=\mu$. (3) $a(z)\equiv1,b(z)\equi

Theorems & Definitions (17)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Theorem 1.2
  • Remark 3
  • Remark 4
  • Example 2.1
  • ...and 7 more