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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.
- Fall '11