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== Homework 5 collaboration area == | == Homework 5 collaboration area == | ||
| − | [[2013 Fall MA 527 Bell|MA527 Fall 2013]] | + | [[2013 Fall MA 527 Bell|MA527 Fall 2013]] |
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| − | Question from [[User:Rrusson|Ryan Russon]] | + | Question from [[User:Rrusson|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. | + | 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 [[User:Bell|Steve Bell]] : | + | Answer from [[User:Bell|Steve Bell]] : |
| − | Those problems don't seem to ask for a sketch, so don't bother. (If you wanted | + | 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.) |
| − | 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 | + | That cryptic remark about the trig identity can be used like so |
| − | <math>-\cos(\frac{\pi}{2}+ x)=\sin x = x -\frac{1}{3!}x ^3 +\dots \approx x</math> | + | <math>-\cos(\frac{\pi}{2}+ x)=\sin x = x -\frac{1}{3!}x ^3 +\dots \approx x</math> |
| − | (when x is small) | + | (when x is small) to realize that the linearized system at the critical point (pi/2,0) has a 1 times y<sub>1</sub> 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. |
| − | to realize that the linearized system at the critical point (pi/2,0) | + | |
| − | has a 1 times y<sub>1</sub> 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 [[User:Rrusson|Ryan Russon]] | + | Response from [[User:Rrusson|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. | Thanks Steve! Using the Jacobian is how I approached it and it is a lot more intuitive for me to approach it that way. | ||
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| − | Question from Christine | + | 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! | + | 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 | + | 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). | + | 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. |
| − | --[[User:Dalec|Dalec]] 21:45, 22 September 2013 (UTC) | + | --[[User:Dalec|Dalec]] 21:45, 22 September 2013 (UTC) |
| + | '''Response from Mickey Rhoades [[User:Mrhoade|Mrhoade]]''' | ||
| − | + | If you follow the method Dalec suggested, you will find that y<sub>1</sub> = -y<sub>2</sub> from the second equation and that if you sub this into the first equation, your critical points are at (0,0) and (-2,+2). Then, linearize with the jacobian and sub in each critical point into the Jacobian matrix and find the eigenvalues. You should obtain two eigenvectors from each critical point, and using the rules from class you should obtain the types of critical points. - Mick | |
| − | + | <br> Question from Craig | |
| − | + | Regarding #11 on p.151 | |
| − | + | When setting up my equation, I am getting the matrix {[0,1],[-2,-2]}. Working through the problem, this gives me the complex e-values of -1+i and -1-i, which corresponds to the stable spiral answer in the back of the book. However, when I go to calculate the e-vectors, I am not getting an empty bottom row for either e-value. Herein lies my confusion. This non-zero bottom row is causing for two bound variables, and a complex e-vector [-1-i,2]. | |
| − | + | Have I missed a step, or set the problem up incorrectly? Any insight would be appreciated. | |
| + | --[[User:Czehrung|Czehrung]] 17:42, 23 September 2013 | ||
| − | [[ | + | <br> [[2013 Fall MA 527 Bell|Back to MA527, Fall 2013]] |
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| + | [[Category:MA527Fall2013Bell]] [[Category:MA527]] [[Category:Math]] | ||
Revision as of 11:46, 23 September 2013
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)
Response from Mickey Rhoades Mrhoade
If you follow the method Dalec suggested, you will find that y1 = -y2 from the second equation and that if you sub this into the first equation, your critical points are at (0,0) and (-2,+2). Then, linearize with the jacobian and sub in each critical point into the Jacobian matrix and find the eigenvalues. You should obtain two eigenvectors from each critical point, and using the rules from class you should obtain the types of critical points. - Mick
Question from Craig
Regarding #11 on p.151
When setting up my equation, I am getting the matrix {[0,1],[-2,-2]}. Working through the problem, this gives me the complex e-values of -1+i and -1-i, which corresponds to the stable spiral answer in the back of the book. However, when I go to calculate the e-vectors, I am not getting an empty bottom row for either e-value. Herein lies my confusion. This non-zero bottom row is causing for two bound variables, and a complex e-vector [-1-i,2].
Have I missed a step, or set the problem up incorrectly? Any insight would be appreciated.
--Czehrung 17:42, 23 September 2013
