application of the linear differential equations on the plane and elements of nonlinear systems, in economics

in recent years, it has become increasingly important to incorporate explicit dynamics in economic analysis. these two tools that mathematicians have developed, differential equations and optimal control theory, are probably the most basic for economists to analyze dynamic problems. in this paper i will consider the linear differential equations on the plane (phase diagram) and elements of nonlinear systems, when we have unequal real roots of the same signs and opposite signs of characteristic roots, and the applications of the theory of differential equations to certain macroeconomic problems. the basic tools for discussion are phase diagram techniques.