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Unformatted text preview: Systems of first-order differential equations 143 Our initial value problem is, then x = f ( x , y , t ) y = g ( x , y , t ) x ( t ) = x , y ( t ) = y (4.2) Economic models invariably involve both linear and nonlinear systems of equa- tions that are autonomous. It is, therefore, worth exploring the meaning of au- tonomous systems in more detail because it is this characteristic that allows much of the graphical analysis we observe in economic theory. In order to elaborate on the ideas we need to develop, consider an extremely simple set of differential equations. Example 4.1 x = 2 x y = y x ( t ) = 2 , y ( t ) = 3 (4.3) We can capture the movement of the system in the following way. Construct a plane in terms of x and y . Then the initial point is ( x , y ) = (2,3). The movement of the system away from this initial point is indicated by the systems of motion, or transition functions, x ( t ) and y ( t ). If we can solve the system for x ( t ) and y ( t ), then we can plot the path of the system in the (...
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- Fall '11