General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find; although, powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases particular solutions may be found with little effort if it is possible to factorize the equation in terms of first-order differential operators. In our present study, we use this factorization technique to address the problem of finding solutions of a system of general two-coupled Liénard-type nonlinear differential equations. We describe a generic algorithm to identify specific classes of Liénard-type systems for which solutions may be found. We demonstrate this method by identifying a class of two-coupled equations for which the particular solution can be found by solving a Bernoulli equation. This class of equations include coupled generalization of the modified Emden equation. We further deduce the general solution of a class of coupled ODEs using the factorization procedure discussed in this paper.

1.
P.
Painlevé
,
Acta Math.
25
,
1
(
1902
).
2.
V. K.
Chandrasekar
,
M.
Senthilvelan
, and
M.
Lakshmanan
,
J. Phys. A: Math. Theor.
40
,
4717
(
2007
).
3.
I. C.
Moreira
,
Hadronic. J.
7
,
475
(
1984
).
4.
P. G. L.
Leach
,
J. Math. Phys.
26
,
2510
(
1985
).
5.
S.
Chandrasekhar
,
An Introduction to the Study of Stellar Structure
(
Dover
,
New York
,
1957
).
6.
J. M.
Dixon
and
J. A.
Tuszynski
,
Phys. Rev. A
41
,
4166
(
1990
).
7.
V. J.
Erwin
,
W. F.
Ames
, and
E.
Adams
,
Wave Phenomena: Modern Theory and Applications
(
North-Holland
,
Amsterdam
,
1984
).
8.
G. C.
McVittie
,
Mon. Not. R. Astron. Soc.
93
,
325
(
1933
).
9.
G. C.
McVittie
,
Ann. Inst. Henri Poincaré
6
,
1
(
1967
).
10.
G. C.
McVittie
,
Ann. Inst. Henri Poincaré
3
,
231
(
1984
).
11.
J. S.R.
Chisholm
and
A. K.
Common
,
J. Phys. A
20
,
5459
(
1987
).
12.
C. N.
Yang
and
R. L.
Mills
,
Phys. Rev.
96
,
191
(
1954
).
13.
V. V.
Golubev
,
Lectures on Analytical Theory of Differential Equations
(
Gostekhizdat
,
Moscow
,
1950
).
14.
B.
van der Pol
,
Philos. Mag.
3
,
65
(
1927
).
15.
16.
O.
Cornejo Pérez
and
H. C.
Rosu
,
Prog. Theor. Phys.
114
,
533
(
2005
).
17.
O.
Cornejo Pérez
,
J.
Negro
,
L.M
Nieto
, and
H. C.
Rosu
,
Found. Phys.
36
,
1587
(
2006
).
18.
L. M.
Berkovich
,
Appl. Anal. Discrete Math.
1
,
122
(
2007
).
19.
D. S.
Wang
and
H.
Li
,
J. Math. Anal. Appl.
343
,
273
(
2008
).
20.
M. A.
Reyes
and
H. C.
Rosu
,
J. Phys. A: Math. Theor.
41
,
285206
(
2008
).
21.
R.
Gladwin Pradeep
,
V. K.
Chandrasekar
,
M.
Senthilvelan
, and
M.
Lakshmanan
,
J. Phys. A: Math. Theor.
42
,
135206
(
2009
).
22.
V. K.
Chandrasekar
,
M.
Senthilvelan
, and
M.
Lakshmanan
,
Proc. R. Soc. A
465
,
609
(
2009
).
23.
V. K.
Chandrasekar
,
M.
Senthilvelan
, and
M.
Lakshmanan
,
Phys. Rev. E
72
,
066203
(
2005
).
24.
R.
Gladwin Pradeep
,
V. K.
Chandrasekar
,
M.
Senthilvelan
, and
M.
Lakshmanan
,
J. Math. Phys.
51
,
103513
(
2010
).
You do not currently have access to this content.