10 12/04

1. New: 11.3.8 Old: 10.3.8 Classify the critical point. State the type of stability. Draw eigenvectors, or their real and imaginary parts, and solution curves accurately. Use a ruler and measure it. Neatly draw at least two solution curves in every distinguishable region. Put direction arrows on all the curves. Make sure the correct slopes can clearly be distinguished on the solution curves at large positive and negative times.

2. New: 11.3.10 Old: 10.3.10 Classify the critical point. State the type of stability. Draw eigenvectors, or their real and imaginary parts, and solution curves accurately. Use a ruler and measure it. Neatly draw at least two solution curves in every distinguishable region. Put direction arrows on all the curves. Make sure the correct slopes can clearly be distinguished on the solution curves at large positive and negative times.

3. New: 11.5.2 Old: 10.5.2
   1. Find the critical points. One critical point is easy. More critical points can be found numerically. In particular their $y$-values are $\pm1.1107$ and $\pm1.6074$.
   2. Find the matrix of derivatives of vector $\vec F$ at each critical point.
   3. Use it to analyze each critical point. List type of point and its stability. Sketch the solution lines in the immediate vicinity of each critical point.

4. Draw comprehensive solution curves based on the critical points and a grid of local slopes.

5. Compare the picture you got in the previous question quantitatively with the positions of the critical points and the directions of the eigenvectors that you got using critical point analysis. State whether critical point analysis must give the right solution near the point, and whether it does.

For the second last question, you will want to use some computer program to plot or at least print out slopes at say 30 times 30, or 900 points. If you are willing to log onto unix and run a fortran program, a link is here.

A better solution may be to use a direction field program from the web. The one that seems nicest to me is this one. Also found here.

See here for Matlab software. (You will need to convert to an ODE by taking the ratio of the equations, and then the software might crash when it divides by zero if it hits a critical point.)

Another to try is here.

The Windows screen-grabber I use is called Printkey. I am sure there are others.