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The Green's function method which has been originally proposed for linearsystems has several extensions to the case of nonlinear equations. A recentextension has been proposed to deal with certain applications in quantum fieldtheory. The general solution of second order nonlinear differential equationsis represented in terms of a so-called short time expansion. The first term ofthe expansion has been shown to be an efficient approximation of the solutionfor small values of the state variable. The proceeding terms contribute to theerror correction. This paper is devoted to extension of the short timeexpansion solution to non-linearities depending on the first derivative of theunknown function. Under a proper assumption on the nonlinear term, a generalrepresentation for Green's function is derived. It is also shown how theknowledge of nonlinear Green's function can be used to study the spectrum ofthe nonlinear operator. Particular cases and their numerical analysis supportthe advantage of the method. The technique we discuss grants to obtain a closedform analytic solution for non-homogeneous non-linear PDEs so far amenable justto numerical solutions. This opens up the possibility of several applicationsin physics and engineering.

General Representation of Nonlinear Green's Function for Second Order Differential Equations Nonlinear in the First Derivative