Unformatted text preview: 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 . All points ( x , y ) on this iso cline have the same slope m . Obtain the inclination α = arctan m ≤ α ≤ 180 ◦ Along the isocline φ ( x , y ) = m construct line elements with inclination α . (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 andages to solve this differential equation explicitly, and in appendices 2....
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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.
 Fall '11
 Dr.Gwartney
 Economics

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