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The main aim of this work is to find the complete solution of some kinds of linear and nonlinear partial differential equations of the second order with variable coefficients. We used Specific assumptions to find the complete solutions of these kinds of equations. These assumptions transform these kinds of equations to linear and nonlinear first order ordinary differential equations respectively. This work contains four chapters: Chapter one: presents some previous works of researchers in differential equations. Chapter Two:shows some previous works of researchers in differential equations. Chapter Three: presents the method of finding the complete solution of four special kinds linear second order partial differential equations with variable coefficients. Chapter Four:discusses useful assumptions which can be used to find the complete solution of three special kinds of nonlinear second order (PDEs) with variable coefficients which are of homogeneous degree (with degree 2).