Second order Linear Differential Equations of Fuchsian Type with Four Singularities
We study a system of linear singularly perturbed functional differential equations by the method
 of integral manifolds. We construct a change of variables that decomposes this system into two
 subsystems, an ordinary differential equation on the center manifold and integral equation...
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| Дата: | 2001 |
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Інститут математики НАН України
2001
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| Цитувати: | Second order Linear Differential Equations of Fuchsian Type with Four Singularities / N.A. Lukashevich // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 308-315 . — Бібліогр.: 1 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860172903212908544 |
|---|---|
| author | Lukashevich, N.A. |
| author_facet | Lukashevich, N.A. |
| citation_txt | Second order Linear Differential Equations of Fuchsian Type with Four Singularities / N.A. Lukashevich // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 308-315 . — Бібліогр.: 1 назв. — англ. |
| collection | DSpace DC |
| container_title | Нелінійні коливання |
| description | We study a system of linear singularly perturbed functional differential equations by the method
of integral manifolds. We construct a change of variables that decomposes this system into two
subsystems, an ordinary differential equation on the center manifold and integral equations on
the stable manifold.
|
| first_indexed | 2025-12-07T17:59:15Z |
| format | Article |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS OF FUCHSIAN
TYPE WITH FOUR SINGULARITIES
N. A. Lukashevich
Belarus University
pr. F. Skoriny, 4, Minsk, 220050, Belarus
We study a system of linear singularly perturbed functional differential equations by the method
of integral manifolds. We construct a change of variables that decomposes this system into two
subsystems, an ordinary differential equation on the center manifold and integral equations on
the stable manifold.
AMS Subject Classification: 34A30
Consider a second order linear differential equation,
y′′ + p(x)y′ + q(x)y = 0, (1)
where p(x) and q(x) are arbitrary analytic functions. Given the initial conditions x = x0,
y(x0) = y0, y′(x0) = y′0, suppose we know a particular solution of the equation, y1(x). Let
any other solution, which is linearly independent of y1, be given by the formula
y = ξ(x)y1. (2)
By differentiating (2) along the solution y1, we successively find that
2ξ′y′1 + (pξ′ + ξ′′)y1 = 0, (3)
(3ξ′′ − pξ′)y′1 + (pξ′′ + p′ξ′ − 2qξ′ + ξ′′′)y1 = 0. (4)
Eliminating the variable y1(x) and its derivative from equations (3) and (4), we get the Schwarz
equations for determining the function ξ(x),
2ξ′ξ′′′ − 3ξ′′
2
+ (p2 + 2p′ − 4q)ξ′
2
= 0. (5)
By setting
ξ′ = η, η′ = wη (6)
in (5), to find the function w(x), we get the Riccati equation
2w′ = w2 − (p2 + 2p′ − 4q). (7)
It follows from (6) and (7) that, in order to find a general solution of equation (1), it is suffi-
cient to find a particular solution of equation (7). In the sequel, we consider equation (1) as a
308 c© N. A. Lukashevich, 2001
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE.. . 309
Fuchsian type equation with four singularities located in the points x = 0, a1, a2, and in x = ∞
(a1, a2 6= 0, a1 6= a2) and written in the form
y′′ +
p0x
2 + p1x+ p2
x(x− a1)(x− a2)
y′ +
q0x
4 + q1x
3 + q2x
2 + q3x+ q4
x2(x− a1)2(x− a2)2
y = 0. (8)
The constant coefficients pk and qk, k = 0, 4, must have the following form in this case [1]:
p0 = α1 + α2 + α3,
p1 = −(α1a2 + α2a1 + α3(a1 + a2)),
p2 = α3a1a2, αk = 1− ρk1 − ρk2, k = 1, 2, 3,
(9)
and
q0 = β4, q1 = b− (a1 + a2)β4, q2 = β1 + β2 + β3 + a1a2β4 − (a1 + a2)b,
q3 = −β1a2 − β2a1 − β3(a1 + a2) + ba1a2, q4 = β3a1a2,
(10)
β1 = ρ11ρ12a1(a1 − a2), β2 = ρ21ρ22a2(a2 − a1),
β3 = ρ31ρ32a1a2, β4 = ρ01ρ02,
where b is the accessor coefficient and the following Fuchsian condition holds:
3∑
k=0
(1− ρk1 − ρk2) = 2, (11)
where ρ01 and ρ02 are exponents with respect to the point z = ∞.
Let us look for a solution of (7) in the form
w =
v0x
2 + v1x+ v2
x(x− a1)(x− a2)
. (12)
Substituting (12) into (7) we find
2
(
−v0x4 − 2v1x
3 + (v0a1a2 + v1(a1 + a2)− 3v2)x
2
+ 2v2(a1 + a2)x− v2a1a2
)
= (v0x
2 + v1x+ v2)
2 − (p0x
2 + p1x+ p2)
2
+ 4(q0x
4 + q1x
3 + q2x
2 + q3x+ q4)− 2
(
−p0x4 − 2p1x
3
+ (p0a1a2 + p1(a1 + a2)− 3p2)x
2 + 2p2(a1 + a2)x− p2a1a2
)
. (13)
310 N.A. LUKASHEVICH
Using (13) we get the following system for finding the unknowns v0, v1, and v2:
(v0 + 1)2 = p20 − 4q0 + 1− 2p0, (v0 + 2)v1 = (p0 − 2)p1 − 2q1,
2(v0a1a2 + v1(a1 + a2)− 3v2)
= v21 + 2v0v2 − p21 − 2p0p2 + 4q2 − 2(p0a1a2 + p1(a1 + a2)− 3p2),
2v2(a1 + a2) = v1v2 − p1p2 + 3q3 − 2p2(a1 + a2),
v22 + 2v2a1a2 − p22 + 2p2a1a2 + 4q4 = 0.
(14)
Using notations (9), (10) and identity (11) we find from the first equation of system (14) that
v0 = ε1(ρ01 − ρ02)− 1, ε21 = 1. (15)
Similarly, from the fifth equation of system (14) we get
v2 = (ε2(ρ31 − ρ32)− 1)a1a2, ε22 = 1. (16)
The second and the fourth equations of system (14), with the use of (15) and (16), become
(ε1(ρ01 − ρ02) + 1)v1 + 2b = γ11a1 + γ12a2,
(ε2(ρ31 − ρ32)− 1)v1 + 2b = γ21a1 + γ22a2,
(17)
where
γ11 = α0(α2 + α3) + 2β4, γ12 = α0(α1 + α3) + 2β4,
γ21 = 2(ε2(ρ31 − ρ32)− 1) + α3(α0 + α1) + 2(ρ31ρ32 + ρ11ρ12 − ρ21ρ22),
γ22 = 2(ε2(ρ31 − ρ32)− 1) + α3(α0 + α2) + 2(ρ31ρ32 − ρ11ρ12 + ρ21ρ22).
(18)
Using system (17) we find that
[ε1(ρ01 − ρ02)− ε2(ρ31 − ρ32) + 2]v1 = (γ11 − γ21)a1 + (γ12 − γ22)a2
and if
δ ≡ ε1(ρ01 − ρ02)− ε2(ρ31 − ρ32) + 2 6= 0, (19)
then
v1 =
1
δ
[(γ11 − γ21)a1 + (γ12 − γ22)a2], (20)
b =
1
2δ
[(ε1(ρ01 − ρ02) + 1)γ21 − (ε2(ρ31 − ρ32)− 1)γ11]a1
+
1
2δ
[(ε1(ρ01 − ρ02) + 1)γ22 − (ε2(ρ31 − ρ32)− 1)γ12]a2. (21)
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE.. . 311
The third equation of (14) becomes
(v1 − a1 − a2)2 − (p1 + a1 + a2)
2 = 2(a1a2 − v2)v0 − 6v2 + 2p0(p2 + a1a2)− 6p2 − 4q2,
or using notations (9), (10) and identities (11), (20), and (21) we get
k0a
2
1 + 2k1a1a2 + k2a
2
2 = 0, (22)
where
k0 ≡ (γ11 − γ21 − δ)2 − (α2 + α3 − 1)2δ2 + 4ρ11ρ12δ
2
− 2δ[(ε1(ρ01 − ρ02) + 1)γ21 − (ε2(ρ31 − ρ32)− 1)γ11],
k1 ≡ (γ11 − γ12 − δ)(γ12 − γ22 − δ)− (α2 + α3 − 1)(α1 + α3 − 1)δ2
+ 2(ρ31ρ32 + ρ01ρ02 − ρ11ρ12 − ρ21ρ22)δ2 − δ[(ε1(ρ01 − ρ02) + 1)(γ21 + γ22)
− (ε2(ρ31 − ρ32)− 1)(γ11 + γ12)]
− [2δ − ε1ε2(ρ01 − ρ02)(ρ31 − ρ32)− α0α3 − α0 − α3 − 1]δ2, (23)
k2 ≡ (γ12 − γ22 − δ)2 − (α1 + α3 − 1)2δ2
+ 4ρ21ρ22δ
2 − 2δ[(ε1(ρ01 − ρ02) + 1)γ22 − (ε2(ρ31 − ρ32)− 1)γ12].
Equation (22) is a condition imposed on the coefficients of equation (8) so that the function
given by (12) is a partial solution of equation (7). Considering (22) as a quadratic equation for
the unknowns ak, k = 1, 2, we should keep in mind that its roots, λk, k = 1, 2, as follows from
the sense of the problem, must be distinct and nonzero. Suppose we found from (22) that
a1 = λka2, k = 1, 2, λk 6= 1. (24)
Represent the particular solution (12) of the Riccati equation (7) as
v0x
2 + v1x+ v2
x(x− a1)(x− a2)
=
r1
x
+
r2
x− a1
+
r3
x− a2
. (25)
To evaluate the unknowns rk, k = 1, 2, 3, (25) gives the system
r1 + r2 + r3 = ε1(ρ01 − ρ01)− 1,
(r1 + r3)a1 + (r1 + r2)a2 =
1
δ
[(γ21 − γ11)a1 + (γ22 − γ12)a2], (26)
r1 = ε2(ρ31 − ρ32)− 1.
312 N.A. LUKASHEVICH
Using (24) we find from system (26) that
r2 = δ − 2− r3, (27)
where
r3 =
1
λk − 1
[
1
δ
(γ21 − γ11)λk +
1
δ
(γ22 − γ12) + 2− δ
− (1 + λk)(ε2(ρ31 − ρ32)− 1)
]
.
Let us set, in equation (7),
W =
r1
x
+
r2
x− a1
+
r3
x− a2
+ V. (28)
To find the function V , we have the following equation:
2V ′ = V 2 +
(r1
x
+
r2
x− a1
+
r3
x− a2
)
V,
from which we find that
V =
2xr1(x− a1)r2(x− a2)r3
C1 −
∫
xr1(x− a1)r2(x− a2)r3 dx
,
and, consequently,
W =
r1
x
+
r2
x− a1
+
r3
x− a2
+
2xr1(x− a1)r2(x− a2)r3
C1 −
∫
xr1(x− a1)r2(x− a2)r3 dx
. (29)
By substituting (29) into formulas (6), we find
η(x) = C2
xr1(x− a1)r2(x− a2)r3[
C1 −
∫
xr1(x− a1)r2(x− a2)r3 dx
]2 ,
(30)
ξ(x) = C3 + C2
1
−C1 +
∫
xr1(x− a1)r2(x− a2)r3 dx
.
Now, using equation (3) find y1(x). Namely,
y1(x) =
C4
C2
x−
1
2
(r1+α1)(x− a1)−
1
2
(r2+α2)(x− a2)−
1
2
(r3+α3)
×
[
C1 −
∫
xr1(x− a1)r2(x− a2)r3 dx
]
. (31)
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE.. . 313
Substituting (30) and (31) into formula (2) we finally find that
y(x) = ξ(x)y1(x)
= x−
1
2
(r1+α1)(x− a1)−
1
2
(r2+α2)(x− a2)−
1
2
(r3−α3)
×
[
C + C1
∫
xr1(x− a1)r2(x− a2)r3 dx
]
, (32)
where C and C1 are new arbitrary constants.
The preceding gives the following theorem.
Theorem. For equation (8) to have a general solution of the form (32), it is sufficient that 1)
the accessor coefficient b have the form (21) and 2) its coefficients satisfy the condition (22).
Together with equation (8), consider the related Heun equation
y′′ +
(α+ β + 1)x2 − [a(γ + δ) + α+ β − δ + 1]x+ aγ
x(x− 1)(x− a)
y′
+
(αβx− q)
x(x− 1)(x− a)
y = 0, (33)
the coefficients of which, as opposed to the coefficients of (9) and (10), have the form
p0 = α+ β + 1, p1 = −[a(γ + δ) + α+ β − δ + 1], p2 = aγ, a1 = 1, a2 = a, (34)
q0 = αβ, q1 = −(a+ 1)αβ − q, q2 = aαβ + (a+ 1)q, q3 = −aq, q4 = 0. (35)
Using the structure of the general solution of equation (8) in the form (32), a particular solution
of (33) is sought in the form
y1 = xs1(x− 1)s2(x− a)s3 , (36)
where the constants s1, s2, s3 are to be found. From (36) we get
y′ =
(
s1
x
+
s2
x− 1
+
s3
x− a
)
y,
(37)
y′′ =
[(
s1
x
+
s2
x− 1
+
s3
x− a
)2
−
(
s1
x2
+
s2
(x− 1)2
+
s3
(x− a)2
)]
y.
314 N.A. LUKASHEVICH
Substituting (37) into (33) we get the system
(s1 + s2 + s3)
2 + (p0 − 1)(s1 + s2 + s3) + q0 = 0,
2s1(s1 − 1)(a+ 1) + 2as2(s2 − 1) + 2s3(s3 − 1) + 2s1s2(2a+ 1)
+2s2s3(a+ 1) + 2s1s3(a+ 2) + p0[(a+ 1)s1 + as2 + s3]
−p1(s1 + s2 + s3)− q1 = 0,
s1(s1 − 1)(a2 + 4a+ 1) + s2(s2 − 1)a2 + s3(s3 − 1) + 2s1s2(a
2 + 2a)
+2s2s3a+ 2s1s3(1 + 2a) + p0as1 − p1[(a+ 1)s1 + as2 + s3]
+p2(s1 + s2 + s3) + q2 = 0, (38)
2s1(s1 − 1)(a2 + 2a) + 2s1s2a
2 + 2s1s3a− p1as1
+p2[(a+ 1)s1 + as2 + s3]− q3 = 0,
s1(s1 − 1)a2 + p2as1 = 0.
It follows from the first and the fifth equations of system (38) that
1) either s1 + s2 + s3 = −α,
2) or s1 + s2 + s3 = −β and (39)
3) either s1 = 0, 4) or s1 = 1− γ.
The fourth equation of system (38) defines the accessor coefficient q,
q = −[2s1(s1 − 1)(a+ 2) + 2s1s2a+ 2s1s3 − s1p1 + γ((a+ 1)s1 + as2 + s3)]. (40)
Substituting (40) into the second equation of (38) and setting
s2 = h− s1 − s3, (41)
where h equals either −α or −β we find that
(2s1 − 2h+ γ − α− β + 1)(a− 1)s3 = [2(s1 − h)(h− 1) + h(γ − p0)]a
+3s31 − (2h+ 1)s1 + (γ − p0)s1 + p1(h− s1). (42)
Assume that, for any choice of s1 and h, the quantity
2s1 − 2h+ γ − α− β + 1 6= 0. (43)
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS OF FUCHSIAN TYPE.. . 315
Note that, if a 6= 1, then assuming that the condition (43) holds, the quantities s1, s2, and s3
can be uniquely expressed in terms of the parameters α, β, γ, δ, and a using formulas (39),
(41), and (42). Substituting their values into the third equation of system (38), the condition
implies that equation (33) has a particular solution of the form (36). Then the general solution
of equation (33) will be
y = xs1(x− 1)s2(x− a)s3
×
[
C1 + C2
∫
x−2s1(x− 1)−2s2(x− a)−2s3 exp
(
−
∫
p(x) dx
)
dx
]
. (44)
The cases where the condition (19) or (43) is violated and the comparison of general solutions
of the forms (32) and (44) are not considered in this paper.
REFERENCES
1. Golubev V. V. Lectures on Analytic Theory of Differential Equations [in Russian], Moscow; Leningrad (1950).
Received 16.04.2001
|
| id | nasplib_isofts_kiev_ua-123456789-174690 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T17:59:15Z |
| publishDate | 2001 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Lukashevich, N.A. 2021-01-27T10:59:50Z 2021-01-27T10:59:50Z 2001 Second order Linear Differential Equations of Fuchsian Type with Four Singularities / N.A. Lukashevich // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 308-315 . — Бібліогр.: 1 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/174690 We study a system of linear singularly perturbed functional differential equations by the method
 of integral manifolds. We construct a change of variables that decomposes this system into two
 subsystems, an ordinary differential equation on the center manifold and integral equations on
 the stable manifold. en Інститут математики НАН України Нелінійні коливання Second order Linear Differential Equations of Fuchsian Type with Four Singularities Лінійні диференціальні рівняння другого порядку типу Фукса з чотирма особливостями Линейные дифференциальные уравнения второго порядка типа Фукса с четырьмя особенностями Article published earlier |
| spellingShingle | Second order Linear Differential Equations of Fuchsian Type with Four Singularities Lukashevich, N.A. |
| title | Second order Linear Differential Equations of Fuchsian Type with Four Singularities |
| title_alt | Лінійні диференціальні рівняння другого порядку типу Фукса з чотирма особливостями Линейные дифференциальные уравнения второго порядка типа Фукса с четырьмя особенностями |
| title_full | Second order Linear Differential Equations of Fuchsian Type with Four Singularities |
| title_fullStr | Second order Linear Differential Equations of Fuchsian Type with Four Singularities |
| title_full_unstemmed | Second order Linear Differential Equations of Fuchsian Type with Four Singularities |
| title_short | Second order Linear Differential Equations of Fuchsian Type with Four Singularities |
| title_sort | second order linear differential equations of fuchsian type with four singularities |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/174690 |
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