Economics Dynamics Problems 201

Economics Dynamics Problems 201 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 185 Hence, x1 = x0 + f (x0 , y0 ) t = 2 + 3(0.01) = 2.03 y1 = y0 + g(x0 , y0 ) t = 2 + 3(0.01) = 2.03 and f (x1 , y1 ) = −2(2.03) − 2.03 + 9 = 2.91 g(x1 , y1 ) = −2.03 + 2.03 + 3 = 3 giving x2 = x1 + f (x1 , y1 ) t = 2.03 + 2.91(0.01) = 2.0591 y2 = y1 + g(x1 , y1 ) t = 2.03 + 3(0.01) = 2.06 This process is repeated. But all this can readily be set out on a spreadsheet, as shown in figure 4.33. The first two columns are simply the differential equations. Columns (3) and (4) employ the Euler approximation using relative addresses and the absolute address for t. The x-y plot gives the trajectory of the system in the phase plane, with initial value (x0 , y0 ) = (2, 2). As can be seen from the embedded graph in the spreadsheet, this trajectory is the same as that shown in figure 4.11 (p. 155) The advantage of using Euler’s approximation, along with a spreadsheet, is that no explicit solution need be obtained – assuming that one exists. By reducing the step size a smoother trajectory results. It is also easy to increase the number of steps. Figure 4.33. ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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