Homework 5 collaboration area
Question from Ryan Russon
For problems p.159: 4,7,11 are we supposed to accompany each solution with a sketch of the what is happening at the critical points or are we fine just stating what is happening at those points based on the eigenvalues of the linearized system? My confusion is stemming from the answer in the back of the book for #11 which says, "Use -cos(+- 1/2...)," you get the picture.
Answer from Steve Bell :
Those problems don't seem to ask for a sketch, so don't bother. (If you wanted to practice for the exam, testing yourself to see if you would know how to draw a sketch if you had to would be therapeutic.)
That cryptic remark about the trig identity can be used like so
$ -\cos(\frac{\pi}{2}+ x)=\sin x = x -\frac{1}{3!}x ^3 +\dots \approx x $
(when x is small) to realize that the linearized system at the critical point (pi/2,0) has a 1 times y1 in that spot. I think it's easier to use the Jacobian matrix to find the first order Taylor term the way I demonstrated in class to see this.
Response from Ryan Russon
Thanks Steve! Using the Jacobian is how I approached it and it is a lot more intuitive for me to approach it that way.
Question from Christine
For problems p.159: I am looking for some suggestion for starting #7. Factoring with the two variables to get the eigen values is not working out...? Thanks!
Answer from Chris
Once you have the equations for y1' and y2', you can set them both equal to zero, and use algebra to find which combinations of y1 and y2 satisfy y1'=y2'=0 (these are your critical points). Then, linearize your equations for y1' and y2' using the Jacobian method used in class. From there, it shouldn't be hard to find the eigenvalues at each critical point.
--Dalec 21:45, 22 September 2013 (UTC)
