Click here to open a new window with a live diagram that we will use in this section. Arrange these two windows so that you can move easily back-and-forth between them. This diagram is set up to study the system of equations:
or, in matrix form,
with the initial value x = 6 and y = 9 Notice the blue ball at the point (6, 9). This point represents the initial values. Notice the red arrow emanating from this point. It represents the vector (-1.5, -6) that shows the strength and direction of the rate of change described given by the system of differential equations at this point. You can drag the blue ball around to see the strength and direction of change at other points. One way to think about this is that the system of differential equations describes the currents in a lake., At each point the system describes the strength and direction of the current at that point. By dragging the blue ball around you can see the strength and direction of the current at different points.
After you have experimented a bit, drag the blue ball back to the initial value (6, 9). You can explore the solution to the initial value problem
by slowly dragging the blue ball in the direction indicated by the red arrow. As you drag the blue ball you will have to change direction as the direction indicated by the red arrow chjanges.
Questions:
with various different initial values.
with various different initial values.
with various different initial values.