Find the equilibrium solution and eigenvalues and eigenvectors of system?

  • Thread starter Norm850
  • Start date
  • #1
11
0
Hey guys, I need to find the equilibrium solution (critical point) for the given system. Also I need to take the homogeneous equation x' = Ax (matrix notation) and find the eigenvalues and eigenvectors.

system: x' = -x - 4y - 4
y' = x - y - 6


Can you help?

Thanks
 
Last edited:

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,836
251
Hi Norm!!! :smile:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :wink:

(start with the eigenvalues and eigenvectors)
 
  • #3
11
0
Okay so, the critical points are when x' and y' equal zero I believe, so adding the two equations gets -5y - 10 = 0 => y = -2, x = 4. So the critical point is (x,y) = (4,-2).

Now for writing the homogeneous equation in matrix form, by using change of variables:

x_1 = x
x_2 = y
x_2 = y'

Gives equations:

(x_1)' = -x_1 - 4x_2 - 4
(x_2)' = x_1 - x_2 - 6

And that gives the matrix form x' = Ax + b, which would be (follow link)
http://i39.tinypic.com/35858ye.png

but we want the homogeneous, so we have x' = Ax, which would be (follow link)
http://i41.tinypic.com/53r0hi.png

Now to find eigenvalues, det(A - λI) = λ^2 + 2λ + 5.

This is where i'm confused so far. I can do the quadratic but we haven't had to do quadratic equation yet so I want to make sure i have everything correct so far?

Thanks
 
  • #4
tiny-tim
Science Advisor
Homework Helper
25,836
251
Hi Norm850! :smile:

(just got up :zzz:)
… I want to make sure i have everything correct so far?

Fine so far. :smile:
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,847
964
You will get a homogeneous set of equations if you let [itex]x_1= x- 4[/itex] and [itex]x_2= y+ 2[/itex]. Of course, [itex]x_1'= x'[/itex] and [itex]x_2'= y'[/itex] and [itex]x= x_1+ 4[/itex], [itex]y= x_2- 2[/itex]. Putting those into the equation.

[itex]x_1'= -(x_1+ 4)-4(x_2- 2)- 4= -x_1- 4x_2[/itex]
[itex]x_2'= (x_1+ 4)- (x_2- 2)- 6= x_1- x_2[/itex]

Determine the eigenvalues of
[tex]\begin{bmatrix}-1 & 4 \\ 1 & -1\end{bmatrix}[/tex]
 

Related Threads on Find the equilibrium solution and eigenvalues and eigenvectors of system?

Replies
4
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
4
Views
825
Replies
2
Views
1K
Replies
3
Views
6K
Replies
2
Views
34K
  • Last Post
Replies
9
Views
1K
Replies
1
Views
849
Replies
0
Views
2K
  • Last Post
Replies
1
Views
2K
Top