Integration of Shazy equation with constant coefficients
The problem of constructing a general solution for the third order Shazy differential equation with six
 constants parameters is considered. This equation belongs to P-type and is connected with Painleve equations. Розглядається задача побудови загального розв’язку диференцiального рiвняння...
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| Date: | 2003 |
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Інститут математики НАН України
2003
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| Cite this: | Integration of Shazy equation with constant coefficients / A.V. Chichurin // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 133-143. — Бібліогр.: 6 назв. — англ. |
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| author | Chichurin, A.V. |
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| citation_txt | Integration of Shazy equation with constant coefficients / A.V. Chichurin // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 133-143. — Бібліогр.: 6 назв. — англ. |
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| description | The problem of constructing a general solution for the third order Shazy differential equation with six
constants parameters is considered. This equation belongs to P-type and is connected with Painleve equations.
Розглядається задача побудови загального розв’язку диференцiального рiвняння Шазi третього порядку з шiстьома сталими параметрами. Це рiвняння належить до типу P i пов’язане з
рiвнянням Пенлеве
|
| first_indexed | 2025-12-07T16:31:38Z |
| format | Article |
| fulltext |
UDC 517.9
INTEGRATION OF SHAZY EQUATION WITH CONSTANT COEFFICIENTS
IНТЕГРУВАННЯ РIВНЯННЯ ШАЗI IЗ СТАЛИМИ КОЕФIЦIЄНТАМИ
A. V. Chichurin
Brest State Univ.
Kosmonavtov Av., 21, Brest, 224665, Belarus
e-mail: chio@tut.by
The problem of constructing a general solution for the third order Shazy differential equation with six
constants parameters is considered. This equation belongs to P -type and is connected with Painleve equati-
ons.
Розглядається задача побудови загального розв’язку диференцiального рiвняння Шазi третьо-
го порядку з шiстьома сталими параметрами. Це рiвняння належить до типу P i пов’язане з
рiвнянням Пенлеве.
Introduction. Investigating the equations
u′′′ = R(u′′, u′, u, z), (1)
where R is a rational function of u′′, u′, u with coefficients analytic in z, on whether they are of
P -type (solutions of such equations do not have moveable critical singular points), Chazy built
the equation [1]
u′′′ =
6∑
k=1
(u′− a′k)(u′′ − a′′k)+Ak(u′ − a′k)3 +Bk(u′ − a′k)2 + Ck(u′ − a′k)
u− ak
+
+Du′′ + Eu′ +
6∏
k=1
(u− ak)
6∑
i=1
Fk
u− ak
. (2)
The 32 coefficients of equation (2) are functions of z, Ak, Bk, Ck, Fk, D,E, ak, k = 1, 6.
Necessary and sufficient conditions for equation (2) to be of P -type make system (S) [2] that
consists of 31 algebraic and differential equations. The unknown functions in system (S) are the
functions Ak, Bk, Ck, Fk, D,E, ak, k = 1, 6.
Equation (2) is connected quite closely with Painleve equations [3]. Investigation of equati-
on (2) is also connected with the theory of isomonodromy deformation of linear systems, the
theory of golonomic quantum fields, and nonlinear evolution equations.
The coefficients Ak, k = 1, 6, must have the following form [2]:
Ak = − 1
ak
, k = 1, 6. (3)
c© A. V. Chichurin, 2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1 133
134 A.V. CHICHURIN
If the coefficients ak, k = 1, 6, are constants, then the functions Bk, Ck, Fk, k = 1, 6, must
have the following form [4]:
Bk = −1
3
D, Ck =
ak
3
(
E − 1
3
D′ +
2
9
D2
)
,
(4)
Fk =
ak
3φ(ak)
(
1
3
D′′ − 2
3
DD′ − E′ + 2
3
DE +
4
27
D3
)
,
where φ(ak) ≡
∏
(ak− aj), k, j = 1, 6; j 6= k. Functions (4) satisfy system (S). For D = 0 and
E being a constant, equation (2) with coefficients (3) and (4) becomes
u′′′ =
6∑
k=1
u′u′′
u− ak
−
6∑
k=1
u′3
ak(u− ak)
+ E
(
1 +
1
4
6∑
k=1
ak
u− ak
)
u′. (5)
By setting
du
dz
= η, η2 = y (6)
in (5), we obtain the linear equation
d2y
du2
=
6∑
k=1
1
u− ak
dy
du
− 2
6∑
k=1
y
ak(u− ak)
+ 2E
(
1 +
1
4
6∑
k=1
ak
u− ak
)
. (7)
In [5], a solution of homogeneous linear equation corresponding to equation (7) was built as
a generalized power series. It was also shown that for some conditions imposed on coefficients,
the power series gives the solution in the form of hypergeometric or even elementary functions.
Let us set
6∑
k=1
1
x− ak
=
6x5 + 4σ2x
3 − 3σ3x
2 + 2σ4x
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
,
6∑
k=1
1
ak(x− ak)
=
6x4 + 4σ2x
2 − 3σ3x+ 2σ4
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
,
6∑
k=1
ak
x− ak
=
−2σ2x
4 + 3σ3x
3 − 4σ4x
2 − 6σ6
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
and consider the next linear equation:
y′′ − 6x5 + 4σ2x
3 − 3σ3x
2 + 2σ4x
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
y′ + 2
6x4 + 4σ2x
2 − 3σ3x+ 2σ4
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
y = 0, (8)
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INTEGRATION OF SHAZY EQUATION WITH CONSTANT COEFFICIENTS 135
where σi, i = 2, 3, 4, 6, are elementary symmetric polynomials, consisting of the elements ak,
k = 1, 6 . Here we used the results obtained in [2], σ1 = σ5 = 0, and the notation x ≡ u.
Equation (8) is a homogeneous equation corresponding to equation (7) and belongs to
Fuchsian type with six singularities located in the points x = ai, ai 6= aj , i 6= j, i, j = 1, 6.
Integrating equation (8) we find a solution of equation (7) and then easily integrate equation
(5) using substitution (6).
Problem. The aim of the work is to prove existence of a solution of equation (8). Also, we
consider a method for constructing such a solution.
Let us consider the following procedure [6] for the linear differential equation of the second
order,
y′′ + p(x)y′ + q(x)y = 0, (9)
where p(x) and q(x) are analytic functions of z. Suppose we know a partial solution y1(x) for
certain initial conditions x = x0, y(x0) = y0, y′(x0) = y′0. We’ll consider the partial solution
y = ξ(x)y1 (10)
of equation (1), which is linearly dependent on the solution y1.
If we differentiate equality (10) along solution y1, then we successively find
2ξ′y′1 + (pξ′ + ξ′′)y1 = 0, (11)
(3ξ′′ − pξ′)y′1 + (pξ′′ + p′ξ′ − 2qξ′ + ξ′′′)y1 = 0. (12)
Eliminating the values y1(x), y′1(x) from equations (11), (12) we construct a Shwarz equati-
on for determining the function ξ(x),
2ξ′ξ′′′ − 3ξ′′2 + (p2 + 2p′ − 4q)ξ′2 = 0. (13)
Let us put, in equation (13),
ξ′ = η, η′ = w η. (14)
Then we obtain the Riccati equation for w(x),
2w′ = w2 − (p2 + 2p′ − 4q). (15)
By setting
w = v − p
in (15), we get the equation
2v′ = 4q − 2pv + v2. (16)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
136 A.V. CHICHURIN
It follows from (14) – (16) that, in order to find a general solution of equation (9), it is suffi-
cient to find a particular solution of equation (16).
For equation (16) that corresponds to equation (8), we seek a solution in the form
v =
b0x
5 + b1x
4 + b2x
3 + b3x
2 + b4x+ b5
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
, (17)
where bi, i = 0, 5, σ2, σ3, σ4, σ6 are unknown coefficients. Substituting (17) into (16) we find an
equation of tenth degree in x. Equating the coefficients of this equation to zero we construct a
system of eleven equations, which we denote by (A). The first equation in system (A) is obtained
from the coefficient at the term of tenth degree and has the form
b20 + 14b0 + 48 = 0. (18)
From equation (18), we find
b0 = −8 (19)
or
b0 = −6.
Remark 1. Below we consider b0 = −8, because if b0 = −6, then it follows from system (A)
that bi = 0, i = 1, 5, σ2 = σ3 = σ4 = σ6 = 0. In the case b0 = − 6, solution (17) becomes
v ≡ −6
x
.
The second equation of system (A) becomes
16b1 + 2b0b1 = 0.
Using (19) we get from the last equation that b1 is an arbitrary constant. Let us consider a
system formed by all equations of system (A) from the third to the seventh one, inclusively. We
find a solution of this system for the unknowns b2, b3, b4, b5, σ6,
b2 = −1
2
(b21 + 32σ2),
b3 =
1
4
(b31 + 24b1σ2 + 56σ3),
b4 = − 1
24
(3b41 + 92b21σ2 + 512σ2
2 + 96b1σ3 + 320σ4), (20)
b5 =
1
48
(3b51 + 112b31σ2 + 976b1σ2
2 + 114b21σ3 + 1440σ2σ3 + 160b1σ4),
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INTEGRATION OF SHAZY EQUATION WITH CONSTANT COEFFICIENTS 137
σ6 = − 1
2048
(5b61 + 220b41σ2 + 2672b21σ
2
2 + 6144σ3
2 + 220b31σ3 +
+ 4032b1σ2σ3 + 1728σ2
3 + 304b21σ4 + 4096σ2σ4).
Substitute (20) into four equations of system (A) (this equation is obtained got from the
coefficients of the third-zero power of x). As the result, we have four relations,
27
256
b71 +
377
64
b51σ2 +
369
64
b41σ3 +
677
4
b21σ2σ3 + 32σ3(27σ2
2 + 7σ4) +
+
1
48
b31(4987σ2
2 + 375σ4) +
1
12
b1(6688σ3
2 + 783σ2
3 + 2080σ2σ4) = 0, (21)
7293b81 + 41292b61σ2 + 52812b51σ3 + 1692864b31σ2σ3 +
+ 73728b1σ3(151σ2
2 + 34σ4) + 16b41(46691σ2
2 + 4383σ4) + 64b21(66688σ3
2 +
+ 14823σ2
3 + 27136σ2σ4) + 8192(160σ4
2 + 837σ2σ
2
3 + 416σ2
2σ4 + 256σ2
4) = 0, (22)
243b91 + 16524b71σ2 + 14580b61σ3 + 575712b41σ2σ3 +
+ 25344b21σ3(249σ2
2 + 43σ4) + 16b51(25063σ2
2 + 2133σ4) +
+ 46080σ3(416σ3
2 − 27σ2
3 + 160σ2σ4) + 128b31(31843σ3
2 + 1377σ2
3 + 9151σ2σ4) +
+ 512b1(28640σ4
2 + 2241σ2σ
2
3 + 17056σ2
2σ4 + 2240σ2
4) = 0, (23)
243b10
1 + 18144b81σ2 + 18468b71σ3 + 933264b51σ2σ3 +
+ 2304b31σ3(6607σ2
2 + 322σ4) + 16b61(31022σ2
2 + 1269σ4) +
+ 18432b1σ3(4448σ3
2 − 27σ2
3 + 400σ2σ4) + 61b41(91645σ3
2 + 5265σ2
3 + 10753σ2σ4) +
+ 256b21(96416σ4
2 + 34641σ2σ
2
3 + 16768σ2
2σ4 + 1376σ2
4)−
− 98304(96σ5
2 − 648σ2
2σ
2
3 + 160σ3
2σ4 + 27σ2
3σ4 + 64σ2σ
2
4) = 0. (24)
Below we find a solution of system (21) – (24). We use the resultant to eliminate σ4 from
equations (21), (22) and from equations (23), (24). In the result we have two equations. From
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
138 A.V. CHICHURIN
this two equations, we eliminate σ3. After transformations we obtain the equation
b61(3b21 + 64σ2)5(9b21 + 256 σ2)4P1(b1, σ2)P2(b1, σ2) = 0, (25)
where
P1(b1, σ2) ≡ 30649071407558418 b30
1 + 12114936173440097280 b28
1 σ2 +
+ 2195536741376273255457 b26
1 σ
2
2 + 242545992862227840359064 b24
1 σ3
2 +
+ 18297161691160879588644864 b22
1 σ4
2 + 999655093084027958425843712 b20
1 σ5
2 +
+ 40899809155897515619716431872 b18
1 σ6
2 + 1276939511343798846743623237632 b16
1 σ7
2 +
+ 30689212561899625725713285382144 b14
1 σ8
2 + 567987097532735199786827071356928 b12
1 σ9
2 +
+ 245 · 3 · 2535107 · 30014232359263 b10
1 σ10
2 + 253 · 19 · 12329 · 40392774440657 b81 σ
11
2 +
+ 260 · 3 · 11 · 883 · 19558829049743 b61 σ
12
2 + 269 · 32 · 654105344721161 b41 σ
13
2 +
+ 277 · 33 · 37 · 311 · 601 · 399523 b21 σ
14
2 + 288 · 34 · 53 · 72 · 11 · 17 · 19 · 31 σ15
2 . (26)
Let
λ =
σ2
b21
.
Then the polynomial P1(b1, σ2) is a polynomial of 15-th degree in λ. The polynomial P2(b1, σ2)
is a polynomial of 24-th degree in λ. We don’t show this polynomial for its large size.
Solutions of equation (25) are
b1 = 0
or
σ2 = − 3
64
b21, (27)
or
σ2 = − 9
256
b21,
or roots of the equation
P1(λ) = 0 (28)
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INTEGRATION OF SHAZY EQUATION WITH CONSTANT COEFFICIENTS 139
or
P2(λ) = 0. (29)
Cases (27) and also the case of remark 1, b0 = −6, corresponde to the situation where some
of six poles aj , j = 1, 6, coincide. Taking into account the character of the solved problem we
don’t consider this cases.
Let us consider equation (28). This equation does not have rational roots. Below we write
the roots of this equation with the precision of twenty digits,
λ = −0, 07320121649118275031,
λ = −0, 07082863896615321427∓ 0, 00019613284001135784 I,
λ = −0, 04717955463774,
λ = −0, 04709244625007,
λ = −0, 04686596251523∓ 0, 00016284672972 I,
λ = −0, 04035287961726453∓ 0, 00156250716732476 I, (30)
λ = −0, 03815572721541967,
λ = −0, 036772571395259496,
λ = −0, 03375957925135584779∓ 0, 00599574993519825017 I,
λ = −0, 0262756967073984240412,
λ = −0, 01430691759131807678894385.
The interval where the real parts of the roots (30) are located lies in the interval [−0, 074;
−0, 014].
Let us consider the second case corresponding to equation (29). This equation is an algebraic
equation of 24-th degree in λ. Equation (29) does not have rational roots. Below we write the
roots of this equation with the precision of twenty digits,
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140 A.V. CHICHURIN
λ = −0, 05973182469049811563,
λ = −0, 05924753382551228670∓ 0, 00182156033988359224 I,
λ = −0, 05696256387908262978∓ 0, 00261736440986289830 I,
λ = −0, 05239705906429039678∓ 0, 00537643598666310768 I,
λ = −0, 04747230324687809597∓ 0, 00072094501657168285 I,
λ = −0, 046139329747961699462,
λ = −0, 045129361296322231964,
λ = −0, 04337787890350930667∓ 0, 00035738165795966455 I,
(31)
λ = −0, 03764480589616372658,
λ = −0, 03579279750781801977∓ 0, 00947339513257097077 I,
λ = −0, 02296144667645378004∓ 0, 00835932269884471527 I,
λ = −0, 02226493882167980806,
λ = 0, 00498921439984309221∓ 0, 00598656611957620107 I,
λ = 0, 08137504883311225792,
λ = 0, 19203841474092165564,
λ = 0, 24383263714533381017.
The interval where the real parts of the roots (31) are located is in the interval [−0, 06; 0, 25].
We can calculate 39 roots of equations (30), (31) an with arbitrary precision. So, we can find
partial solutions (17) with an arbitrary precision.
Suppose that the precision of the calculation for solutions of equations (30), (31) is known.
Suppose also that we have found the partial solution (17) of equation (16). To find a general
solution of equation (5) we construct the following procedure.
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INTEGRATION OF SHAZY EQUATION WITH CONSTANT COEFFICIENTS 141
Let us represent the function v(x)− p(x) as a sum of six elementary fractions,
4x5 + b1x
4 + (b2 − 4σ2)x3 + (b3 − 3σ3)x2 + (b4 − 2σ4)x+ b5
(x− a1)(x− a2)(x− a3)(x− a4)(x− a5)(x− a6)
=
6∑
i=1
ki
x− ai
. (32)
To evaluate the unknowns ki, i = 1, 6, from (32), we have the system of equations,
k1 + k2 + k3 + k4 + k5 + k6 = −2,
a1k1 + a2k2 + a3k3 + a4k4 + a5k5 + a6k6 = b1,
a2
1k1 + a2
2k2 + a2
3k3 + a2
4k4 + a2
5k5 + a2
6k6 = b2 − 2σ2,
(33)
a3
1k1 + a3
2k2 + a3
3k3 + a3
4k4 + a3
5k5 + a3
6k6 = b3 − b1σ2 − 5σ3,
a4
1k1 + a4
2k2 + a4
3k3 + a4
4k4 + a4
5k5 + a4
6k6 = b4 + b1σ3 − (b2 + 2σ2)σ2,
a−1
1 k1 + a−1
2 k2 + a−1
3 k3 + a−1
4 k4 + a−1
5 k5 + a−1
6 k6 = −b5/σ6.
Let us consider system (33) with the unknowns k1, k2, k3, k4, k5. Using properties of the
Vandermonde determinant we find a solution of this system in the form
kj =
1
a3
6m
(a4
6aib4 + a4
6ai(a6 + ai)b3 +
+ ai(a5
6ai + a4
6(a2
i − σ2) + a3
6σ3 − a4
6σ4 − σ6)b2 +
+ 2(a5
6ai(aiσ2 − σ3 − a2
6aiσ2σ4 + a4
6ai(a
2
iσ2 − σ2
2 − aiσ3 + σ4) +
+ a3
6(aiσ2σ3 − σ6)− aiσ2σ6)− a6k6(2a7
6ai + 2a6
6a
2
i + 2a5
6a
3
i + a4
6ai(a
3
i + σ3) +
+ a3
6ai(aiσ3 − 2σ4)− 2a6aiσ6 − a2
iσ6 + a2
6(σ6 − a2
iσ4))), j = 1, 5,
k6 =
1
m
(a3
6b5 + a4
6b4 + a5
6b3 − (a4
6σ2 − a3
6σ3 + a2
6σ4 + σ6)b2 −
− a6(a4
6σ2 − a3
6σ3 + a2
6σ4 + σ4)b1 −
− 2(a5
6σ3 + a4
6(σ2
2 − σ4)− a3
6σ2σ3 + a2
6(σ2σ4 − σ6) + σ2σ6)), (34)
m ≡ a2
6(2a2
6 − a6 + a3
6σ3 − 2a2
6σ4 − 3σ6).
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142 A.V. CHICHURIN
Using notations (34) for known values bj , j = 0, 5, σ2, σ3, σ4, σ6 (these values can be
calculated for each of the 39 roots of equations (30), (31) showen above we find ki, i = 1, 6.
Let us set, in equation (15),
W =
6∑
i=1
ki
x− ai
+ V.
To find the function V , we have the following equation:
2V ′ = V 2 + 2
6∑
i=1
ki
x− ai
V
from which we find that
V =
2
∏6
i=1(x− ai)ki
C1 −
∫ ∏6
i=1(x− ai)kidx
,
and, consequently,
W =
6∑
i=1
ki
x− ai
+
2
∏6
i=1(x− ai)ki
C1 −
∫ ∏6
i=1(x− ai)kidx
. (35)
By substituting (35) into formulas (14), we find
η(x) = C2
∏6
i=1(x− ai)ki
[C1 −
∫ ∏6
i=1(x− ai)kidx]2
(36)
and
ξ(x) = C3 + C2
1
−C1 +
∫ ∏6
i=1(x− ai)kidx
, (37)
where C1, C2, C3 are arbitrary constants.
From (11) and (14) we get
y1 = exp
−1
2
w∫
0
(
p(τ) +
η′(τ)
η(τ)
)
dτ
or, accounting for (36), (37), we find
y1(x) =
1
√
η
e−
1
2
F , (38)
where
F ≡
∫
p(w) dw, p(w) = − 6x5 + 4σ2x
3 − 3σ3x
2 + 2σ4x
x6 + σ2x4 − σ3x3 + σ4x2 + σ6
.
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INTEGRATION OF SHAZY EQUATION WITH CONSTANT COEFFICIENTS 143
Substituting (37) and (38) into (10) we find y(x). The preceding gives the following theorem.
Theorem 1. A general solution of equation (8) has the form
y(x) = ξ(x) y1(x),
where ξ(x) is determined by (37) and y1(x) − by (38).
To construct a solution of equation (7), we use the formula
y = ξ(w)y1 + y2
∫
h(w)eF y1dw − y1
∫
h(w)eF y2 dw, (39)
where
y2 = y1
∫
e−F
y1
2
dw, h(w) ≡ 2E
(
1 +
1
4
6∑
k=1
ak
w − ak
)
. (40)
Using substitution (6) we finally obtain the next theorem.
Theorem 2. General integral of equation (5) is∫
dw√
y(w)
= z + C,
where y(w) is determined by (39), (40) and C is an arbitrary constant.
Remark 2. During the calculations we used CAS Mathematica.
The author thanks professor N. A. Lukashevich for a participation in discussing this paper
and his valuable remarks.
1. Chazy J. Acta Math. — 1911. — 34. — P. 317 – 385.
2. Lukashevich N. A. To the theory of Chazy equation // Different. Equat. — 1993. — 29, №2. — P. 353 – 357.
3. Dobrovolskii V. A. Essays of development of analytical theory of differential equations. — Kiev, 1974. —
455 p.
4. Chichurin A. V. About one solution of Shazy system // Vestnic Brest Univ. — 2000. — №6. — P. 27 – 34.
5. Chichurin A. V. About linear equation of second order with six poles // Proc. Second Int. Workshop ”Mathemati-
ca” Syst. in Teach. and Res. (Siedlce, Poland, January 28 – 30, 2000). — Moscow, 2000. — P. 34 – 44.
6. Lukashevich N. A. Second order linear differential equations of Fuchsian type with four singularities // Nonli-
near Oscillations. — 2001. — 4, №3. — P. 306 – 315.
Received 25.06.2002
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 1
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| id | nasplib_isofts_kiev_ua-123456789-176162 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T16:31:38Z |
| publishDate | 2003 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chichurin, A.V. 2021-02-03T19:29:04Z 2021-02-03T19:29:04Z 2003 Integration of Shazy equation with constant coefficients / A.V. Chichurin // Нелінійні коливання. — 2003. — Т. 6, № 1. — С. 133-143. — Бібліогр.: 6 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/176162 517.9 The problem of constructing a general solution for the third order Shazy differential equation with six
 constants parameters is considered. This equation belongs to P-type and is connected with Painleve equations. Розглядається задача побудови загального розв’язку диференцiального рiвняння Шазi третього порядку з шiстьома сталими параметрами. Це рiвняння належить до типу P i пов’язане з
 рiвнянням Пенлеве en Інститут математики НАН України Нелінійні коливання Integration of Shazy equation with constant coefficients Інтегрування рівняння Шазі із сталими коефіцієнтами Интегрирование уравнения Шази с постоянными коэффициентами Article published earlier |
| spellingShingle | Integration of Shazy equation with constant coefficients Chichurin, A.V. |
| title | Integration of Shazy equation with constant coefficients |
| title_alt | Інтегрування рівняння Шазі із сталими коефіцієнтами Интегрирование уравнения Шази с постоянными коэффициентами |
| title_full | Integration of Shazy equation with constant coefficients |
| title_fullStr | Integration of Shazy equation with constant coefficients |
| title_full_unstemmed | Integration of Shazy equation with constant coefficients |
| title_short | Integration of Shazy equation with constant coefficients |
| title_sort | integration of shazy equation with constant coefficients |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/176162 |
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