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methods for investigation of qualitative properties of solutions to nonlinear
differential equations. Note that Emden – Fowler equation appears for
the first time in [1]. Its physical origin is also described in [2]. This
equation was investigated in detail in the books [3, 4], and later in [5].
See also [6, 7] and references. Asymptotic properties of solutions to
different generalizations of this equation were investigated in [8–35]. The
results concerning asymptotic behavior of solutions to nonlinear ordinary
differential equations is used to describe the properties of solutions to
nonlinear partial differential equations. See, for example, [36–40].
In this article the asymptotic classification of all possible solutions to
the fourth order Emden – Fowler type differential equations
y
IV
(
x
) +
p
0
|
y
|
k
−
1
y
(
x
) = 0
, k >
1
, p
0
>
0
(1)
and
y
IV
(
x
)
−
p
0
|
y
|
k
−
1
y
(
x
) = 0
, k >
1
, p
0
>
0
(2)
is given.
The asymptotic classification of all possible solutions to the third order
Emden – Fowler type differential equations
y
III
(
x
) +
p
(
x
)
|
y
|
k
−
1
y
(
x
) = 0
, k >
1
, p
(
x
)
>
0
(3)
is described.
For fourth-order nonlinear equations, the oscillatory problem was
investigated in [10, 13, 14, 17, 21, 28, 29, 31, 33, 35], in linear case —
in [41].
Phase Sphere.
Note that if a function
y
(
x
)
is a solution to equation
(1), the same is true for the function
z
(
x
) =
Ay
(
Bx
+
C
)
,
(4)
where
A
6
= 0
,
B >
0
,
and
C
are any constants satisfying
|
A
|
k
−
1
=
B
4
.
(5)
Indeed, we have
z
IV
(
x
) +
p
0
|
z
|
k
−
1
z
(
x
)=
AB
4
y
IV
(
Bx
+
C
)+
+
p
0
|
Ay
(
Bx
+
C
)
|
k
−
1
Ay
(
Bx
+
C
) =
=
Ay
IV
(
Bx
+
C
)
B
4
− |
A
|
k
−
1
= 0
.
Any non-trivial solution
y
(
x
)
to equation (1) generates a curve
(
y
(
x
)
, y
0
(
x
)
, y
00
(
x
)
, y
000
(
x
))
in
R
4
\{
0
}
.
Let us introduce in
R
4
\{
0
}
an
equivalence relation such that two solutions connected by (4), (5) generate
equivalent curves, i.e. the curves passing through equivalent points (may
be for different
x
).
4
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2