Direction Field Examples

>	with(DEtools):

Direction field for dy/dx = x^2 - y (this you should have done in Math 305). Here y is a function of x, so wewrite y(x), while x is an indepentdent variable.

>	dfieldplot(diff(y(x),x) = x^2 - y(x) , y(x) , x=-5..5, y=-5..5 , arrows=line);

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Now we plot a direction field for a 2x2 system. Note , the system is autonomus but nonlinear. We will see that there are two fixed points, points where the <x', y'> = <0,0>. One is at the origin. Try to find the location of the other fixed point.

>	dfieldplot([diff(x(t),t)=x(t)*(1-y(t)), diff(y(t),t)=.3*y(t)*(x(t)-1)], [x(t),y(t)],t=-2..2, x=-1..2, y=-1..2);

Next we redo the field above, but plot a true vector field. That is the lengths of the vector are not al set to 1, but they are scaled so that the largest vector has length one..

>	with(plots):

Warning, the name changecoords has been redefined

>	fieldplot([x*(1-y), .3*y*(x-1)], x=-1..2, y=-1..2);

Below I plotted the true vector field, tat is, without any scaling. The vector is in the form [< a,b >, <c,d>], that is te base is at <a,b> and the head is at <c,d>. The indices i and j represent 1/4 inch grid points, hence x = i/4 and y=j/4.

>	arrow([seq(seq([<i/4,j/4>,<(i/4)*(1-j/4),0.3*(j/4)*(i/4-1)>] ,i=-4..8),j=-4..8)], shape=arrow);

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