EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5564
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Investigation of a Fourth-Order Nonlinear Differential
Equation with Moving Singular Points of Algebraic

Type

Magomedyusuf Gasanov

Moscow State University of Civil Engineering, Russia

Abstract. In this study, a fourth-order nonlinear ordinary differential equation is considered.
The specificity of nonlinearity lies in the presence of moving singular points, which hinders the
application of classical theory that only works in the linear case. Two research problems are
addressed in this work: the theorem of existence and uniqueness of the solution, and the precise
criteria for the existence of a moving singular point. These problems are solved in both the real
and complex domains. The specificity of transitioning to the complex plane is demonstrated using
phase spaces. The obtained results are validated through numerical experiments, confirming the
reliability of the results.

2020 Mathematics Subject Classifications: 34A34, 34A12, 34M04, 34M05

Key Words and Phrases: Cauchy problem, movable singular point, analytical approximate
solution, phase spaces, Puiseux series, meromorphic function

1. Introduction

Nonlinear differential equations are widely used in science and engineering, for instance,
in problems of hydrodynamics [8], and continuum mechanics [15, 23].

In this study, we will focus on the Cauchy problem for a fourth-order differential
equation. The equation under consideration in present research has the following form:

d4w

dz4
+Q0w

(
w′)2 = f (z) , (1)

This model can be viewed as a mechanical system involving an oscillator, such as a
mass-spring system or a pendulum, where the oscillator’s motion is influenced by external
forces. The term w (w′)2 represents the interaction between the displacement w and
the square of the oscillator’s velocity w′. It captures the effect of velocity-dependent
forces, which may introduce additional nonlinear behaviors, such as dissipation or energy
generation. Often, when solving many problems, nonlinear terms are neglected. However,

DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5564

Email address: vonasag6991@mail.ru (M. Gasanov)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 2 of 19

as shown in studies [4–6, 9, 10, 12, 14, 17, 22, 25, 44, 48], accounting for nonlinearity in
the form of the derivative squared can impact the solution of the investigated problems.

The complexity of investigating such a class of equations lies in their nonlinearity,
which is a condition for the emergence of movable singular points. The presence of mov-
able singular points of algebraic type poses a challenge for finding an analytical solution.
movable singular points make nonlinear differential equations in general unsolvable in
quadratures. The classification and characteristics of movable and fixed singular points
are well-described in the works [11, 18].

Singular points of the integrals of differential equations, whose position does not depend
on the initial data that determine these integrals, are called fixed singular points .

Singular points of the integrals of differential equations, whose position depends on
the initial data, are called movable singular points.

Algebraic movable singular points include simple and multiple poles, as well as branch
points of finite order. In the neighborhood of such points, a Puiseux series expansion is
assumed.

Despite the presence of movable singular points, there are methods for studying such
equations. The process of finding a solution and investigating its behavior can be divided
into two domains: the domain of analyticity and the neighborhood of the movable singular
point.

In the domain of analyticity, linearization is possible, and numerical methods can be
applied to solve such problems. In [21], two algorithms are developed to check the possibil-
ity of reducing a nonlinear differential equation to a linear one using Lie algebra and point
transformations. In [16], linearization is based on the method of new approximation, tak-
ing into account both local and global properties obtained through global approximation
of Lie derivatives. In [43], the linearization problem was solved using a generalized lin-
earizing transformation. In [41, 42, 45, 46], the authors used homotopy analysis methods
and numerical methods.

The second method for solving this type of nonlinear differential equations is analytical.
Finding an analytical solution is rare due to the complexity of the considered physical
phenomenon [49]. As shown in the works [7, 13, 19, 24, 26, 38, 39, 47], these problems are
solved using a specific variable substitution or special functions.

The third method is asymptotic. The main task of this method is to investigate the
behavior of the solution near singular points (both moving and stationary) as well as at
infinity [1, 2, 50].

The author’s method for studying nonlinear differential equations is based on solving
four mathematical problems: the theorem of existence and uniqueness (in the domain of
analyticity and the neighborhood of the movable singular point); the influence of perturba-
tion of initial data (movable singular point) on the structure of the analytical approximate
solution; precise criteria for the existence of movable singular points; precise boundaries
for the application of the analytical approximate solution. Solving these problems allows
for the development of an algorithm to find a movable singular point with a predetermined
accuracy and to combine the obtained results with existing numerical methods [35]. This
method has already been successfully tested in solving certain classes of equations.



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 3 of 19

In the works [27–30], the described method has been used to investigate the van der
Pol equation in the complex domain. In [29], an approximate analytical solution to the
initial value problem in the domain of analyticity is found. The work [30] is dedicated to
studying the influence of perturbations in initial data on the structure of the approximate
analytical solution. In the article [28], the authors found an approximate analytical solu-
tion in the neighborhood of the movable singular point, while [27] addressed the problem
of the influence of perturbations in the movable singular point on the structure of the
approximate analytical solution.

In [31–36], the authors have successfully solved all research problems related to third-
order nonlinear differential equations with polynomial right-hand side of both second and
seventh degrees.

For other classes of equations, the same method for finding analytical approximate
solutions has been explored in [3, 20, 37, 40].

For the equation (1), two problems are addressed: the classical problem of the theory of
differential equations, the proof of the existence and uniqueness theorem, and the precise
criteria for the existence of a moving singular point. The existence theorem is considered in
the complex domain. The solution is sought in the form of a Puiseux series. Estimates for
the series coefficients are provided to determine the convergence region of the analytic part
of the series. An analytical approximate solution is obtained, along with estimates of its
error. The second problem, finding precise criteria for the existence of a moving singular
point, which has been solved in both real and complex domains due to the different
approaches to solving these problems.

2. Main results

2.1. The theorem of existence and uniqueness. Modification
of the Cauchy–Kovalevskaya theorem.

Let’s consider the following Cauchy problem:

w(4) +Q0w
(
w′)2 = f (z) , (2)

w (z0) = w0,
w′ (z0) = w′

0,
w′′ (z0) = w′′

0 ,
w′′′ (z0) = w′′′

0 ,

(3)

where w′
0, w

′′
0 , w

′′′
0 , Q0 ∈ C.

Theorem 1. Let z∗ be a moving singular point of the Cauchy problem (2) — (3), and let
the function f (z) be holomorphic in the domain |z∗ − z| < ρ1, then the solution w (z) is
representable in the form of a meromorphic function:

w (z) = (z∗ − z)−1
∑
n≥0

An (z
∗ − z)n , (4)



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 4 of 19

in the domain |z − z∗| < ρ2. Here

ρ2 = max

{
ρ1,

1
5
√
|γ|

}
,

where

|γ| = max

{
sup
n

(∣∣∣∣∣f (n) (z∗)

n!

∣∣∣∣∣
)
, |w0| ,

∣∣w′
0

∣∣ , ∣∣w′′
0

∣∣ , ∣∣w′′′
0

∣∣} .

Proof.
We will seek the solution in the form of a generalized power series:

w (z) =
∑
n≥0

An (z
∗ − z)n+r . (5)

Let’s substitute the formula (5) into the equation (2), and obtain:

∑
n≥0

An (n+ r) (n+ r − 1) (n+ r − 2) (n+ r − 3) (z − z∗)n+r−4

+Q0

∑
n≥0

An (z
∗ − z)n+r

∑
n≥0

An (n+ r) (z∗ − z)n+r−1

2

=
∑
n≥0

Dn (z
∗ − z)n .

After cubing the first term on the right-hand side, we get:

∑
n≥0

An (n+ r) (n+ r − 1) (n+ r − 2) (n+ r − 3) (z∗ − z)n+r−4

= −Q0

∑
n≥0

An (z
∗ − z)n+r

∑
n≥0

Ã∗
n (z

∗ − z)n+2r−2


+
∑
n≥0

Dn (z
∗ − z)n ,

∑
n≥0

An (n+ r) (n+ r − 1) (n+ r − 2) (n+ r − 3) (z∗ − z)n+r−4

= −Q0

∑
n≥0

C∗
n (z

∗ − z)n+3r−2 +
∑
n≥0

Dn (z
∗ − z)n ,



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 5 of 19

where 
A∗

n =
∑n

i=0AiAn−i,

C∗
n =

∑n
i=0 Ã

∗
iAn−i,

Ã∗
n =

∑n
i=0BiBn−i,

Bn = An (n+ r) .

The left and right sides are identically equal, from which the following conditions
follow:


n+ r − 4 = n+ 3r − 2
An (n+ r) (n+ r − 1) (n+ r − 2) (n+ r − 3) = −Q0C

∗
n, if n = 0, 1, 2, 3, 4

An (n+ r) (n+ r − 1) (n+ r − 2) (n+ r − 3) = −Q0C
∗
n +Dn−5, if n ≥ 5

(6)

From the first equality, it follows that r = −1. As shown in [11], if r ∈ Q−, then z∗ is
a movable singular point for solving the Cauchy problem (2) — (3).

The second and third equalities are recurrent relations that allow for the unique de-
termination of the coefficients of the series (2.1).

Writing the initial values for n, we obtain the first few recurrent relations:

A0 =
√
− 24

Q0

A1 = A2 = A3 = A4 = 0

A5 =
D0
192

A6 =
D1
336

A7 =
D2
1152

. . . . . . . . .

Taking into account the regularity, it is evident that:
An = Dn−5

C −A0 · F (D0Dn−10, D1Dn−9, . . . , DlDk) , l + k = n− 10, C = const, n ≥ 5.

Since the series is formal, it is necessary to find its convergence domain. To do this, we
will use the modified majorant method used in the Cauchy–Kovalevskaya theorem. This
method is based on the estimation of the coefficients An on the basis of which the majority
series is constructed.

Theorem 2. Assume that all conditions of Theorem 1 be satisfied; then the estimate for
the coefficients of the series expansion (2.1), for sufficiently large values of n, takes the
following form:

|An| ≤
|γ|[

n
5 ]
(
|Q0A0|

∣∣[n−9
2

]
+ 1
∣∣+ 1

)
|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|

, (7)

where [α] denotes the integer part of the number.

Proof. From the system (2.1), it follows that for sufficiently large values of the index,
the following equality can be used:



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 6 of 19

An (n− 1) (n− 2) (n− 3) (n− 4) = −Q0C
∗
n +Dn−5. (8)

Let’s expand the right-hand side:

−Q0

n∑
i=0

Ã∗
iAn−i +Dn−5 = −Q0

n∑
i=0

 i∑
j=0

BjBi−j

An−i +Dn−5

= −Q0

n∑
i=0

 i∑
j=0

Aj (j − 1)Ai−j (i− j − 1)

An−i +Dn−5

= −Q0

(
n∑

i=0

(− (i− 1)A0Ai + . . .− (i− 1)AiA0)An−i

)
+Dn−5

= −Q0

(
−AnA

2
0 − 2 (n− 1)AnA

2
0 + P (A1, . . . , An−1)

)
+Dn−5.

Considering the obtained equality, equation (8) can be expressed as:

An (n− 1) (n− 2) (n− 3) (n− 4)

= −Q0

(
−AnA

2
0 − 2 (n− 1)AnA

2
0 + P (A0, . . . , An−1)

)
+Dn−5,

An ((n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3))

= −Q0 · P (A0, . . . , An−1) +Dn−5,

An =
Dn−5 −Q0 · P (A0, . . . , An−1)

(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)
.

Considering the form of the function P (A0, . . . , An−1) and the initial values Ai, as well
as the condition of analyticity of the function f (z), which implies the boundedness of the

coefficients |Dn| =
∣∣∣f (n)(x0)

n!

∣∣∣ ≤ θn, we obtain the following estimate for the coefficients:

|An| =
∣∣∣∣ Dn−5 −Q0 · P (A0, . . . , An−1)

(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)

∣∣∣∣ ≤
≤ |Dn−5|+ |Q0 · P (A0, . . . , An−1)|

|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|
≤



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 7 of 19

≤

|Dn−5|+

∣∣∣∣∣∣∣∣Q0A0

∑
k+l=n−9
k,l∈N∪{0}

DkDl

∣∣∣∣∣∣∣∣
|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|

≤

≤
|γ|[

n
5 ] +

∣∣∣Q0A0γ
[n5 ]
∣∣∣ ∣∣[n−9

2

]
+ 1
∣∣

|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|
=

=
|γ|[

n
5 ]
(
|Q0A0|

∣∣[n−9
2

]
+ 1
∣∣+ 1

)
|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|

.

Let’s find the radius of convergence of the analytic part of the series (2.1), taking into
account the obtained estimate (7):

|z∗ − z|5 < 1

|γ|
⇒ |z∗ − z| < 1

5
√
|γ|

.

Taking into account the existing estimates for the series coefficients, we can write the
analytical approximate solution:

wN (z) = (z∗ − z)−1
N∑

n=0

An (z
∗ − z)n , (9)

in the domain |z∗ − z| < ρ2.
Next, let’s proceed to estimate the error of the analytical approximate solution.

Theorem 3. Assume that all conditions of Theorem 1 and Theorem 2 be satisfied, then
the analytical approximate solution (9) of the Cauchy problem (2) — (3), for sufficiently
large values of N , has the following error estimates:

∆wN ≤
4∑

k=0

|γ|N+1+k
(
|Q0A0|

∣∣∣[5(N+1)+k−9
2

]
+ 1
∣∣∣+ 1

)
|z∗ − z|N+k∣∣∣∏4

i=1 (5 (N + 1) + k − i) + 24 ((2 (5 (N + 1) + k)− 3))
∣∣∣

× 1

1− |γ · (z∗ − z)|5
. (10)

Proof. Let’s use the triangle inequality:

∆wN =

∣∣∣∣∣∣
∑
n≥0

An (z
∗ − z)n−1 −

N∑
n=0

An (z
∗ − z)n−1

∣∣∣∣∣∣ =
∣∣∣∣∣∣
∑

n≥N+1

An (z
∗ − z)n−1

∣∣∣∣∣∣



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 8 of 19

Considering that |
∑

ai| ≤
∑

|ai| , ∀ai ∈ C, we have:

∆wN =

∣∣∣∣∣∣
∑

n≥N+1

An (z
∗ − z)n−1

∣∣∣∣∣∣ ≤
∑

n≥N+1

|An| |z∗ − z|n−1

≤
∑

n≥N+1

|γ|[
n
5 ]
(
|Q0A0|

∣∣[n−9
2

]
+ 1
∣∣+ 1

)
|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|

|z∗ − z|n−1

≤
∑

n≥N+1

4∑
k=0

|γ|n+k (|Q0A0|
∣∣[5n+k−9

2

]
+ 1
∣∣+ 1

)∣∣∣∏4
i=1 (5n+ k − i) + 24 (2 (5n+ k)− 3)

∣∣∣ |z∗ − z|n+k−1

≤
4∑

k=0

|γ|(N+1)+k
(
|Q0A0|

∣∣∣[5(N+1)+k−9
2

]
+ 1
∣∣∣+ 1

)
|z∗ − z|5(N+1)+k−1∣∣∣∏4

i=1 (5 (N + 1) + k − i) + 24 (2 (5 (N + 1) + k)− 3)
∣∣∣

× 1

1− |γ · (z∗ − z)|5
.

2.2. Numerical experiment

Let’s consider the Cauchy problem (2) — (3) with a specific example, when Q0 = −4
and f (z) = sin (z) with given initial conditions:

w(4) − 4w
(
w′)2 = sin z (11)

w (0) = 0.2
w′ (0) = 0.3
w′′ (0) = 0
w′′′ (0) = 0

(12)

The coefficients of the series expansion (2.1) for the solution to the Cauchy problem
(11) — (12) take the following form:

A0 =
√
6

A1 = A2 = A3 = A4 = 0
A5 =

1
192

A6 = − 1
2016

A7 =
1

40320
. . . . . . . . .



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 9 of 19

The estimate for the coefficients of the series expansion (2.1) according to Theorem 2
for the Cauchy problem (11) — (12) is given by:

|An| ≤
4
√
6
∣∣[n−9

2

]
+ 1
∣∣+ 1

|(n− 1) (n− 2) (n− 3) (n− 4) + 24 (2n− 3)|
.

Next, according to Theorem 3, we determine the error estimate of the analytical ap-
proximate solution:

∆wN ≤
4∑

k=0

(
4
√
6
∣∣∣[5(N+1)+k−9

2

]
+ 1
∣∣∣+ 1

)
|z∗ − z|5(N+1)+k−1∣∣∣∏4

i=1 (5 (N + 1) + k − i) + 24 (2 (5 (N + 1)) + k − 3)
∣∣∣ 1

1− |z∗ − z|5
.

The value of the moving singular point is z∗ = 3.513, and the convergence radius is
ρ = 1. For a solution error of ε ≈ 10−7 at the point z1 = 3.2, according to Theorem 3, it
is sufficient to take N = 7. In this case, the analytical approximate solution is given by:

w7 (z) =
√
6 (3.513− z)−1 +

1

192
(3.513− z)4 − 1

2016
(3.513− z)5 +

1

40320
(3.513− z)6 .

Below is a table of characteristics and a graph comparing the solution of the Cauchy
problem (11) — (12) obtained using the analytical approximate method proposed by the
authors and the solution obtained numerically using the Runge-Kutta method.

Table 1. Numerical characteristics of the analytical approximate solution.

z1 w7(z1) ∆1

3.2 7.6125622 10−7

2.3. Exact criteria for the existence of a moving singular point in the real
and complex domains

This section is dedicated to criteria for the existence of a moving singular point. Both
point and interval criteria are considered. Point criteria only guarantee the existence of a
moving singular point, while interval criteria allow determining their location. To define
such criteria in the complex plane, a transition to phase spaces is necessary.

Before formulating the theorems, it is necessary to transform the Cauchy problem (2)
— (3) into the inverse Cauchy problem by a change of variable w (z) = 1

u(z) :

w′ = −u−2u′,

w′′ = 2u−3
(
u′
)2 − u−2u′′,

w′′′ = −6
(
u′
)3

u−4 + 6u′u′′u−3 − u′′′u−2,

w(4) = 24
(
u′
)4

u−5 − 36
(
u′
)2

u′′u−4 + 6
(
u′′
)2

u−3 + 3u′u′′′u−3 − u(4)u−2.



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 10 of 19

Figure 1: The solution to the Cauchy problem (11) — (12) the analytical approximate method (dashed blue
line) and the Runge-Kutta method (solid red line)

Thus, the inverse Cauchy problem will have the form:

− u(4)u3 + 33u′u′′′u2 + 6
(
u′′
)2

u2 − 36
(
u′
)2

u′′u

+ 24
(
u′
)4 −Q0u

′u2 − u5f (z) = 0 (13)
u (z0) = u0,
u′ (z0) = u′0,
u′′ (z0) = u′′0,
u′′′ (z0) = u′′′0 .

(14)

Since (13) — (14) is the inverse Cauchy problem, the solution u (z) will take a regular
value at the point z∗, namely, it will be equal to zero.

Let’s proceed to formulate the criteria for the existence of a moving singular point. To
begin with, let’s formulate them for the real domain.

The following lemma helps to take into account the specifics of the regularization
method when constructing an algorithm for finding a moving singular point.

Lemma 1. Let the function be u (z) does not change the sign on some segment [a, b].
Then in order for w (z) to reach a local maximum at point c ∈ (a, b), it is necessary and
sufficient that at this point u (z) had a local minimum.



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 11 of 19

Proof. The proof obviously follows from classical analysis, a necessary and sufficient
condition for a local extremum.

Theorem 4. If z∗ is a movable singular point of the Cauchy problem (2) — (3), and the
function w (z) defined on the half-interval [z0; z

∗) . Then there is a number γ, such that
the function w (z) on the half-interval [γ; z∗) has the following property:[

w (z) , w′ (z) , w′′ (z) , w′′′ (z) > 0,
w (z) , w′ (z) , w′′ (z) , w′′′ (z) < 0.

Proof.
Consider formula (2.1) obtained in Theorem 1. According to the existence theorem

∃ξ : ξ ∈ [z0; z
∗) such that the analytic part of (2.1) converges in the area [ξ; z∗) , then we

get:

w (z) =

√
− 24

Q0
(z∗ − z)−1 +

D0

192
(z∗ − z)4 +

D1

336
(z∗ − z)5 + . . .

Since
1√
Q0

(z∗ − z)−1 z→z∗−0→ +∞,

and (
D0

192
(z∗ − z)4 +

D1

336
(z∗ − z)5 + . . .

)
z→z∗−0→ 0,

there is such a point ξ1 ≥ ξ : ∀z ∈ [ξ1; z
∗) the condition w > 0 is fulfilled.

Similarly, in the case of the derivative:

w′ (z) =

√
− 24

Q0
(z∗ − z)−2 +

D0

48
(z∗ − z)3 +

5D1

336
(z∗ − z)4 + . . .

Since √
− 24

Q0
(z∗ − z)−2 z→z∗−0→ +∞,

and (
D0

48
(z∗ − z)3 +

5D1

336
(z∗ − z)4 + . . .

)
→ 0,

there is such a point ξ1 ≥ ξ : ∀z ∈ [ξ1; z
∗) the condition w′ > 0 is fulfilled. Similarly for

w′′, w′′′.

Theorem 5. Let z (u) be the inverse function to the solution of the inverse Cauchy problem

(13) — (14), and the following conditions hold: z (0) = z∗, z′ (0) = −
√
−Q0

24 , z
′′ (0) =

z′′′ (0) = 0, then z∗ is a moving singular point of the Cauchy problem (2) — (3). This
condition is necessary and sufficient.



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 12 of 19

Proof. Necessity. Let z∗ be a moving singular point of the Cauchy problem (2) — (3),
then the previously proven theorems 1, 2, 3 hold. With the substitution w (z) = 1

u(z) , it is

evident that u (z∗) = 0. Using this substitution to transition to the inverse equation, we
get:

u (z) =
1

(z∗ − z)−1∑
n≥0An (z∗ − z)n

=
∑
n≥0

Ãn (z
∗ − z)n+1 , (15)

where Ã0 =
1
A0

, Ã1 = Ã2 = Ã3 = Ã4 = 0.
Based on the theorem on the inversion of series [11], we obtain the following equality:

z∗ − z (u) =
∑
n≥0

Bnu
n+1, B0 =

1

A0
, B1 = 0, B2 = 0. (16)

From equality (16), it is evident that z (0) = z∗. Further, by differentiating (16), we
get:

z′ (u) = −
(
B0u+B3u

4 + . . .
)′

= −B0 − 4B3u
3 − . . . (17)

From (17), it follows that z′ (0) = −B0 = − 1
A0

= −
√

−Q0

24 . Differentiating (17), we

obtain:
z′′ (u) =

(
−B0 − 4B3u

3 − . . .
)′

= −12B3u
2 − . . . (18)

From (18), we get that z′′ (0) = 0. Following a similar algorithm as before, we obtain
z′′′ (0) = 0.

Sufficiency. Let z (u) be the inverse function to the solution of the inverse Cauchy

problem (13) — (14), and the following conditions hold: z (0) = z∗, z′ (0) = −
√

−Q0

24 ,

z′′ (0) = z′′′ (0) = 0. Let’s prove that z∗ is a moving singular point of the Cauchy problem
(2) — (3).

Since u (z) is an analytic function in the vicinity of the point z∗, its inverse is also
analytic in this domain and can be expanded into a Taylor series:

z (u) =
∑
n≥0

Dnu
n (19)

Given that z (0) = z∗ and the existing expansion (19), we obtain D0 = z∗. Further,
by differentiating equality (19), we get:

z′ (u) =
∑
n≥1

nDnu
n−1,

thus, taking into account the condition z′ (0) = −
√

−Q0

24 , we obtain D1 = −
√

−Q0

24 .

Similarly, we find D2 = D3 = 0. Thus, equality (19) takes the form:

z (u) = z∗ −
√

−Q0

24
u+D4u

4 + . . . ,



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 13 of 19

z∗ − z (u) =

√
−Q0

24
u−D4u

4 + . . . (20)

Based on the theorem on the inversion of series [11], we obtain:

u (z) =

√
−Q0

24
(z∗ − z)− D̃4 (z

∗ − z)4 + . . . (21)

Taking into account the relationship between the solutions of the Cauchy problems (2)
— (3) and (13) — (14), we have:

w (z) =
1

u (z)
=

1√
−Q0

24 (z∗ − z) + . . .

=

√
− 24

Q0
(z∗ − z)−1 + C1 (z

∗ − z) + C2 (z
∗ − z)3 + . . . ,

Thus, z∗ is a moving singular point of algebraic type of the Cauchy problem (2) — (3).

Theorem 6. The fact that z∗ is a moving singular point of the function w (z) is equivalent
to the existence of a certain neighborhood of this point in which the function u (z) is
continuous and has different signs at the endpoints of this interval.

Proof. Necessity. Given that u (z) is the inverse function, then u (z∗) = 0, and
the function is continuous in some neighborhood of this point. Considering that u (z) =√
−Q0

24 (z∗ − z)+o (z∗ − z), when crossing the moving singular point, the function changes

its sign. Thus, there exists an interval on the ends of which the function takes values of
different signs.

Sufficiency. Due to the continuity of the function u (z) and the different signs at
the ends of a certain interval, according to the Bolzano-Cauchy theorem ∃ ξ : u (ξ) = 0.
Taking into account the relation w (z) = 1

u(z) , we obtain that ξ is a moving singular point

of the solution to the Cauchy problem (2) — (3).

Next, let’s proceed to the formulation of the exact criteria for the existence of a moving
singular point in the complex domain. For the complex domain, these criteria are related
to the specifics of transitioning to phase spaces.

Let’s express the solution to the inverse Cauchy problem (13) — (14) as u (z) =
P (x, y)+iQ (x, y) where the functions P (x, y) and Q (x, y) are characterized, respectively,
by the phase spaces Φ1 (x, y, P (x, y)) and Φ2 (x, y,Q (x, y)).

Let’s use the terminology introduced in the work [32] to define correct and incorrect
lines to facilitate the statement of the following theorems.

Theorem 7. For z∗ to be a moving singular point of algebraic type of the solution w (z)
of the Cauchy problem (2) — (3), it is necessary and sufficient that for Re (u (z)) and
Im (u (z)), where the function u (z) is the solution to the inverse Cauchy problem (5) —
(6), in some region G, which is the neighborhood of the regular point z∗ (x∗, y∗) of the
function u (z), the phase spaces Φ1 and Φ2 satisfy the following conditions:



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 14 of 19

(i) P (x, y) and Q (x, y) are continuous with respect to their arguments;

(ii) when crossing the point z∗ (x∗, y∗), moving along the regular line l in the direction
of the axes Ox and Oy, the functions P (x, y) and Q (x, y) change signs, where
l : {z∗ ∈ l ⊂ G, l ∈ (Φ1 ∪ Φ2)} .

Proof. Necessity. According to the theorem statement, we have that z∗ is a moving
singular point of y (z) of the Cauchy problem (2) — (3). Let’s demonstrate that in this
case Re (w (z)) and Im (w (z)) satisfy Theorem 7.

Since Theorem 1 holds, the principal part of the series representing the solution to

the Cauchy problem (2) — (3) takes the form w (z) = O
(
A0/z∗ − z

)
, Therefore, for the

inverse solution in the region G, we can assert that u (z) = o
(
z∗ − z/A0

)
. In this situation,

we have the following:

sgn (P (x, y)) = sgn ((x∗ − x)) (22)

sgn (Q (x, y)) = sgn ((y∗ − y)) . (23)

Analysis of the analytic part u (z): the sign of the function P (x, y) is determined by
the sign of x, while the sign of the function Q (x, y) is determined by the sign of y.

Without loss of generality, let’s assume that the moving singular point z∗ is located
in the first quadrant of the phase plane. As a correct line, we can consider a segment
of the circle |z| = |z∗| in the region G. Moving along this circle in the direction of the

correct line l, we observe that for points z ∈ l : arg z < arg z∗ ⇒
{

x > x∗

y < y∗
and for

points z ∈ l : arg z > arg z∗ ⇒
{

x < x∗

y > y∗
, therefore, the imaginary and real parts of the

function u (z) are continuous functions with respect to the arguments, and they change
their sign when passing through the point z∗. Since, without loss of generality, only the
first quadrant was considered, the necessity is similarly proven in the other quadrants.

Theorem 8. In order for z∗ to be a moving singular point of the function w (z), which is a
solution to the Cauchy problem (2) — (3), it is necessary and sufficient for the imaginary
and real parts of u (z) in some sufficiently small neighborhood G of the point z∗ in the
phase spaces Φ1 and Φ2, to be continuous functions with respect to their arguments and
to change their signs when passing through the point z∗ (x∗, y∗), moving sequentially along
certain incorrect lines l1, l2 : {z∗ ∈ l1 ⊂ G, z∗ ∈ l2 ⊂ G , l1 ∈ Φ1, l2 ∈ Φ2} .

Proof. Necessity. Similar to Theorem 7, we have the principal part of the series,

which is the solution to the Cauchy problem (2) — (3): w (z) = O
(
A0/z∗ − z

)
, and the

analytic part of the inverse function u (z) = o
(
z∗ − z/A0

)
.

Let’s consider the line l1 : y = y∗ = const — an incorrect line with respect to the Ox
axis. Moving along this line, taking into account the signs of the arguments (9) and the



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 15 of 19

theorem of existence and uniqueness of the solution, u (z) as a function of a single variable
changes sign when crossing the point z∗.

Similarly, we consider the line l2 : x = x∗ = const, which is an incorrect line with
respect to the Oy axis, which completes the proof of the necessity.

Sufficiency. Based on the theorem, we conclude that the imaginary and real parts
of the function u (z) are continuous functions with respect to their arguments in some
neighborhood of the point z∗ in the phase spaces Φ1 and Φ2, and change their signs
when crossing the point z∗ (x∗, y∗), moving sequentially along incorrect lines l1 and l2 in
the direction of the corresponding axes l1, l2 : {z∗ ∈ l1 ⊂ G, z∗ ∈ l2 ⊂ G , l1 ∈ Φ1, l2 ∈ Φ2}.
Utilizing this fact, we obtain that u (z∗) = 0, and the function w (z) = P (x, y) + iQ (x, y)
can be represented as:

u (z) =
∑
n≥0

Ãn (z
∗ − z)n+1 .

Taking into account the introduced substitution u (z) = 1
w(z) , we get

w (z) = (z∗ − z)−1
∑
n≥0

An (z
∗ − z)n .

3. Conclusion

In this paper, we considered a nonlinear fourth-order differential equation with a mov-
able singular point of algebraic type. The author proposed an analytical approximate
solution method based on splitting the solution search into the area of analyticity and
the neighborhood of a moving singular point. In this paper, the theorem of existence and
uniqueness in the vicinity of a moving singular point was formulated and proved. The
solution obtained during the proof has a simple pole at the point z∗. Using the modified
majorant method, estimates for the coefficients were obtained, and as a result, the con-
vergence domain of the solution under consideration. The second task of the study was to
formulate the necessary, as well as necessary and sufficient conditions for the existence of
a mobile singular point in both the real and complex domains. The main idea of solving
this problem was the regularization of a moving singular point. The theoretical results
were tested in a numerical experiment. The results of the numerical experiment were
compared with existing numerical methods. The analytical approximate method used by
the author can be applied to other classes of equations, as it was previously indicated in
the introduction. This method complements the existing methods for solving nonlinear
differential equations, such as asymptotic, exact methods and others. This work can be
utilized to formulate an algorithm for finding a movable singular point with a specified
accuracy, by combining the obtained results with numerical methods [35].



M. Gasanov / Eur. J. Pure Appl. Math, 18 (1) (2025), 5564 16 of 19

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