We consider an ordinary differential equation (ODE) of order
n
n
, which is a polynomial of the independent variable
x
x
, the dependent variable
y
y
and all its derivatives up to the order
n
n
. To such equation we put in correspondence its Newton polygon. If the polygon has a vertex
(
v
,
1
)
(v,1)
and corresponding truncated equation has eigenvalues
λ
1
\lambda _1
, …,
λ
n
\lambda _n
, then there exists such formal substitution
y
=
z
+
φ
(
x
)
y=z+\varphi (x)
, where
φ
(
x
)
\varphi (x)
is a power series, that for
z
=
z
′
=
⋯
=
z
(
n
)
=
0
z=z’=\dotsb =z^{(n)} =0
the transformed equation has only resonant terms
a
m
x
m
a_mx^m
, where
m
=
v
+
λ
k
∈
Z
m=v+\lambda _k\in \mathbb Z
. It is true near each of two points:
x
=
0
x=0
and
x
=
∞
x=\infty
.