Abstract: In order to solve the problem of the Green’s theory in the linear time-varying system, this paper studies the Green’s fuinction for solving differential equations and proposes a formular corresponding to the Green’s equation for partial differential equations. Two algorithms based on the Green’s function for solving the differential equation are summarized, including the eigen-value based algorithm and the fundamental algoithm. Their equivalence for both algorithms are also discussed. Furthermore, two methods to determine the initial displacement for a free string vibration equation are presented, i.e. the Green’s function based string displacement method and the eigen-value of the differential equation besed Green’s function method. These results are very important in theory and applications for the development of the theory on the linear time-varying system and non-stationary signal processing.