Economics Dynamics Problems 211

Economics Dynamics Problems 211 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations found since 2e2t 2e2t 2 x = = 2t = 2 t )2 y (3e 9e 9 Hence y= 9x 2 Whether or not it is possible to readily find a Cartesian representation of the parametric curve, it is a simple matter to plot the parametric curve itself using software packages. Example 4.1 with Mathematica The two commands used in this set of instructions, DSolve and ParametricPlot are now both contained in the main package:11 Clear[x,y] sol=DSolve[{x’ [t]==2x[t],y’ [t]==y[t],x[0]==2, y[0]==3}, {x[t],y[t]},t] solx=sol[[1,1,2]] soly=sol[[1,2,2]] x[t-]:=solx y[t-]:=soly traj=ParametricPlot[{x[t],y[t]},{t,0,1}] If the equations for x(t) and y(t) are already known, then only the last instruction need be given. For example, if it is known that x(t) = 2e2t and y(t) = 3et then all that is required is traj=ParametricPlot[{2e2t ,3et },{t,0,1}] Example 4.1 with Maple To use Maple’s routine for plotting parametric equations that are solutions to differential equations it is necessary to load the plots package first. The following input instructions will produce the trajectory for example 4.1: restart; with(plots): sys:={diff(x(t),t)=2*x(t),diff(y(t),t)=y(t), x(0)=2,y(0)=3} vars:={x(t),y(t)}: sol:=dsolve(sys,vars,numeric); odeplot(sol,[x(t),y(t)],0..1,labels=[x,y]); 11 In earlier versions, DSolve and ParametricPlot needed to be loaded first since these were contained in the additional packages. This is no longer necessary, since both are contained in the basic built in functions. 195 ...
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