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Economics Dynamics Problems 60

# Economics Dynamics Problems 60 - 44 Economic Dynamics We...

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44 Economic Dynamics We can summarise the method of isoclines as follows: (1) From the differential equation dy dx = φ ( x , y ) determine the family of isoclines φ ( x , y ) = m and construct several members of this family. (2) Consider a particular isocline φ ( x , y ) = m 0 . All points ( x , y ) on this iso- cline have the same slope m 0 . Obtain the inclination α 0 = arctan m 0 0 α 0 180 Along the isocline φ ( x , y ) = m 0 construct line elements with inclination α 0 . (This establishes part of the direction field.) (3) Repeat step 2 for each isocline. (4) Draw smooth curves to represent the approximate integral curves indi- cated by the line elements of step 3. It is apparent that this is a very tedious procedure. Luckily, a number of mathe- matical software packages now compute direction fields and can be used to con- struct isoclines (see appendices 2.1 and 2.2). Example 2.12 dy dx = 2 x y In sections 2.11 and 2.12 we give the instructions on using software pack- ages to solve this differential equation explicitly, and in appendices 2.1 and 2.2 we provide Mathematica and Maple instructions, respectively, for plot-
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