Find the equilibrium solution and eigenvalues and eigenvectors of system?

• Norm850
In summary, the conversation is about finding the equilibrium solution (critical point) and eigenvalues and eigenvectors for a given system. The critical point is determined to be (4,-2) and the homogeneous equation is rewritten in matrix form. The eigenvalues are then found to be λ^2 + 2λ + 5.
Norm850
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:
Hi Norm!

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

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

Hi Norm850!

(just got up :zzz:)
Norm850 said:
… I want to make sure i have everything correct so far?

Fine so far.

You will get a homogeneous set of equations if you let $x_1= x- 4$ and $x_2= y+ 2$. Of course, $x_1'= x'$ and $x_2'= y'$ and $x= x_1+ 4$, $y= x_2- 2$. Putting those into the equation.

$x_1'= -(x_1+ 4)-4(x_2- 2)- 4= -x_1- 4x_2$
$x_2'= (x_1+ 4)- (x_2- 2)- 6= x_1- x_2$

Determine the eigenvalues of
$$\begin{bmatrix}-1 & 4 \\ 1 & -1\end{bmatrix}$$

1. What is an equilibrium solution in a system?

An equilibrium solution in a system is a state where the system's variables do not change over time. In other words, the system is at rest and there is no net change or movement in the system.

2. How do you find the equilibrium solution of a system?

To find the equilibrium solution of a system, you need to set all of the system's variables to zero and solve the resulting equations. The values obtained for the variables at this point will be the equilibrium solution.

3. What are eigenvalues and eigenvectors in a system?

Eigenvalues and eigenvectors are mathematical concepts used to analyze the behavior of a system. Eigenvalues represent the values of the system's variables at equilibrium, while eigenvectors represent the direction and magnitude of change in the system.

4. How do you calculate eigenvalues and eigenvectors of a system?

To calculate the eigenvalues and eigenvectors of a system, you need to set up a matrix equation and solve for its characteristic polynomial. The roots of the polynomial will be the eigenvalues, and the corresponding eigenvectors can be calculated using these values.

5. What is the significance of eigenvalues and eigenvectors in a system?

Eigenvalues and eigenvectors provide important information about the stability and behavior of a system. They can determine whether a system will reach an equilibrium, and if so, what that equilibrium will look like. They also help in predicting how the system will respond to external influences or perturbations.

• Differential Equations
Replies
2
Views
2K
• Differential Equations
Replies
4
Views
2K
• Linear and Abstract Algebra
Replies
5
Views
353
• Differential Equations
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
793
• Differential Equations
Replies
17
Views
3K
• Differential Equations
Replies
3
Views
1K
• Differential Equations
Replies
13
Views
2K
• Differential Equations
Replies
17
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
788