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Unformatted text preview: Systems of first-order differential equations 145 initial conditions. Of course, with different initial conditions, there will be differ- ent trajectories in the ( x , y )-plane, but these too will be unique for a given initial condition. 4.2 The phase plane, fixed points and stability In chapter 2 we introduced the phase line . This was the plot of x ( t ) on the x-line. It is apparent that figure 4.1(a) is a generalisation of this to two variables. In figure 4.1(a) we have plotted the path of the two variables x and y . At any point in time we have a point such as ( x ( t ) , y ( t )), and since the solution path is uniquely defined for some initial condition ( x , y ), then there is only one path, one function y = φ ( x ), which satisfies the condition y = φ ( x ). A solution curve for two variables is illustrated in figure 4.2(a), whose coor- dinates are ( x ( t ) , y ( t )) as t varies over the solution interval. This curve is called a trajectory, path or orbit of the system; and the (...
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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