Entire solutions of two certain Fermat-type ordinary differential equations

• DOI: 10.1515/math-2023-0120
• Publication Date: 2023-10-11T13:22:47Z
• Authors: Binbin Hu; Liu Yang
• Keywords: polynomial; 30d35; QA1-939; 30d20; entire solution; 39b32; ordinary differential equation; 34m05; Mathematics

Abstract In this article, we investigate the precise expression forms of entire solutions for two certain Fermat-type ordinary differential equations:

(

a

0

f +

a

1

f

′

)

2

+

(

a

0

f +

a

2

f

′

)

2

= p

{\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}=p

and

(

a

0

f +

a

1

f

′

)

2

+

(

a

0

f +

a

2

f

′

)

2

=

e

g

\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

a

0

{a}_{0}

,

a

1

{a}_{1}

, and

a

2

{a}_{2}

are the complex numbers with

a

0

≠ 0

{a}_{0}\ne 0

and

∣

a

1

∣ + ∣

a

2

∣ ≠ 0

| {a}_{1}| +| {a}_{2}| \ne 0

, while

p

p

and

g

g

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.