Economics Dynamics Problems 205

Economics Dynamics Problems 205 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 189 So long as solutions exist, then the packages will solve the system of equations. Thus, the system of three equations with initial values: x (t) = x(t) y (t) = x(t) + 3y(t) − z(t) z (t) = 2y(t) + 3x(t) x(0) = 1, y(0) = 1, z(0) = 2 (4.36) can be solved in a similar manner with no difficulty. In the case of nonlinear systems of differential equations, or where no explicit solution can be found, then it is possible to use the NDSolve command in Mathematica and the dsolve(. . . , numeric) command in Maple to obtain numerical approximations to the solutions. These can then be plotted. But often more information can be obtained from direction field diagrams and phase portraits. A direction field shows a series of small arrows that are tangent vectors to solutions of the system of differential equations. These highlight possible fixed points and most especially the flow of the system over the plane. A phase portrait, on the other hand, is a sample of trajectories (solution curves) for a given system. Figure 4.36(a) shows a direction field and figure 4.36(b) a phase portrait. In many instances direction fields and phase portraits are combined on the one diagram – as we have done in many diagrams in this chapter. The phase portrait can be derived by solving a system of differential equations, if a solution exists. Where no known solution exists, trajectories can be obtained by using numerical Figure 4.36. ...
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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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