  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 Li\'enard type nonlinear
differential equations. We describe a generic algorithm to identify specific
classes of Li\'enard 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 ordinary
differential equations using the factorization procedure discussed in this
manuscript.
