Abstract
AbstractIn this paper, we discuss the growth of meromorphic solutions of some classes of homogeneous and nonhomogeneous differential equations. We will prove that if meromorphic functions $$F, A_j, D_j$$
F
,
A
j
,
D
j
and polynomials $$P_j$$
P
j
with degree $$n\ge 1$$
n
≥
1
$$(j=0,1,\cdots ,k-1)$$
(
j
=
0
,
1
,
⋯
,
k
-
1
)
satisfy some conditions, then the equation $$f^{(k)}+(A_{k-1}e^{P_{k-1}}+D_{k-1})f^{(k-1)}+\cdots +(A_1e^{P_1}+D_1)f'+(A_0e^{P_0}+D_0)f=F(z) (k\ge 2)$$
f
(
k
)
+
(
A
k
-
1
e
P
k
-
1
+
D
k
-
1
)
f
(
k
-
1
)
+
⋯
+
(
A
1
e
P
1
+
D
1
)
f
′
+
(
A
0
e
P
0
+
D
0
)
f
=
F
(
z
)
(
k
≥
2
)
when $$F\equiv 0$$
F
≡
0
, all solutions $$f\not \equiv 0$$
f
≢
0
have infinite order and hyper order $$\sigma _2(f)\ge n$$
σ
2
(
f
)
≥
n
. This is a continue work of [Gan and Sun in Adv. Math. 36(1), 51–60 (2007)] and [Chen and Xu in Electron. J. Qual. Theory Differ. Equ. 1: 1–13 (2009)]
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference14 articles.
1. Chen, Z.X.: The growth of solutions of the differential equation $$f +e^{-z}f +Q(z)f =0$$. Sci. China Ser. A 31, 777–784 (2001). (In Chinese)
2. Chen, Z.X.: On the hyper order of solutions of higher order differential equations. Chin. Ann. Math. Ser. B 24, 501–508 (2003). (In Chinese)
3. Chen, Z.X.: On the growth of solutions of a class of higher order differential equations. Acta Math. Sci. Ser. B 24, 52–60 (2004). (In Chinese)
4. Chen, Z.X.: On the hyper order of solutions of some second order liner differential equations. Acta Math. Sinica B 18, 79–88 (2002). (In Chinese)
5. Chen, Z.X., Shon, K.H.: On the growth and fixed points of solutions of second order differnetial equation with meromorphic coefficients. Acta Math Sinica, English Series 21, 753–764 (2004)