Abstract
A singular nonlinear differential equation
z
σ
d
w
d
z
=
a
w
+
z
w
f
(
z
,
w
)
,
{z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w),
where
σ
>
1
\sigma \gt 1
, is considered in a neighbourhood of the point
z
=
0
z=0
located either in the complex plane
C
{\mathbb{C}}
if
σ
\sigma
is a natural number, in a Riemann surface of a rational function if
σ
\sigma
is a rational number, or in the Riemann surface of logarithmic function if
σ
\sigma
is an irrational number. It is assumed that
w
=
w
(
z
)
w=w\left(z)
,
a
∈
C
⧹
{
0
}
a\in {\mathbb{C}}\setminus \left\{0\right\}
, and that the function
f
f
is analytic in a neighbourhood of the origin in
C
×
C
{\mathbb{C}}\times {\mathbb{C}}
. Considering
σ
\sigma
to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions
w
=
w
(
z
)
w=w\left(z)
in a domain that is part of a neighbourhood of the point
z
=
0
z=0
in
C
{\mathbb{C}}
or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property
lim
z
→
0
w
(
z
)
=
0
{\mathrm{lim}}_{z\to 0}w\left(z)=0
is proved and an asymptotic behaviour of
w
(
z
)
w\left(z)
is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.