Calc III Lab: Centers

Adjust the parameters in DsTool so that we will compute solutions to the linear differential equation determined by
dx/dt = -y
dy/dt = 0.5 x
This means changing the parameters in the Selected Point Panel to read (a,b,c,d)=(0,-1,0.5,0).

Compute a phase portrait for the linear system given above. This means that you must compute trajectories (starting at different initial conditions) until you have a good idea of what every trajectory looks like. For example, for the system above, the phase portrait looks like Figure 1 below.

Figure 1

Activity:

• Compute the trace and determinant of the matrix for the specific values of (a,b,c,d) used above.
• Plot the ordered pair (det, trace) on the diagram provided.
• Next to the ordered pair you just plotted, sketch a small version of the phase portrait associated with the linear ODE determined by this matrix. You may want to put arrows on the phase portrait to indicate the direction that trajectories flow.
• Are there straight-line solutions to this system? If it appears so, indicate these on your sketch of the phase portrait. If not, write "No straight-line solutions" next to the sketch.

• Clear the old phase portrait.
• Adjust the parameters in DsTool so that we will compute solutions to the linear differential equation determined by the matrix with entries (a,b,c,d)=(0,-1,0.125,0).
• Complete the steps specified above to plot the trace and determinant on the supplied diagram, and to sketch the phase portrait with any straight-line solutions of the associated linear ODE.

Go To

• Next page: Foci: Swirling in or out
• Outline Page: Outline
• DsTool page: DsTool help page