  The Green's function method which has been originally proposed for linear
systems has several extensions to the case of nonlinear equations. A recent
extension has been proposed to deal with certain applications in quantum field
theory. The general solution of second order nonlinear differential equations
is represented in terms of a so-called short time expansion. The first term of
the expansion has been shown to be an efficient approximation of the solution
for small values of the state variable. The proceeding terms contribute to the
error correction. This paper is devoted to extension of the short time
expansion solution to non-linearities depending on the first derivative of the
unknown function. Under a proper assumption on the nonlinear term, a general
representation for Green's function is derived. It is also shown how the
knowledge of nonlinear Green's function can be used to study the spectrum of
the nonlinear operator. Particular cases and their numerical analysis support
the advantage of the method. The technique we discuss grants to obtain a closed
form analytic solution for non-homogeneous non-linear PDEs so far amenable just
to numerical solutions. This opens up the possibility of several applications
in physics and engineering.
