Affiliation:
1. Department of Applied Mathematics, Anhui University of Technology , Maanshan, 243032 , P. R. China
Abstract
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.
Reference22 articles.
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2. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
3. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.
4. P. Montel, Lecons sur les familles normales de fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927, pp. 135–136.
5. F. Gross, On the equation fn+gn=1. I, Bull. Amer. Math. Soc. 72 (1966), 86–88.