Entire solutions of two certain Fermat-type ordinary differential equations

Abstract

In this article, we investigate the precise expression forms of entire solutions for two certain Fermat-type ordinary differential equations: (a0f+a1f′)2+(a0f+a2f′)2=p{\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}=p and (a0f+a1f′)2+(a0f+a2f′)2=eg\hspace{0.39em}{\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}={e}^{g} in C{\mathbb{C}} by making use of the Nevanlinna theory for meromorphic functions, where a0{a}_{0}, a1{a}_{1}, and a2{a}_{2} are the complex numbers with a0≠0{a}_{0}\ne 0 and ∣a1∣+∣a2∣≠0| {a}_{1}| +| {a}_{2}| \ne 0, while pp and gg are the polynomials in C{\mathbb{C}}. Moreover, some examples are given to illustrate the existence of entire solutions for the aforementioned two certain equations.

Keywords

• entire solution
• ordinary differential equation
• polynomial
• 30d20
• 30d35
• 34m05
• 39b32