System of Differential Equations, Phase Plane

In summary, the conversation discussed problem #1, a-c, which involves the main equations dx/dt=Ax, (A-rI)v=0, and det(A-rI)=0. The attempt at a solution showed some confidence in the answer to A and B, but difficulty in starting C. The presence of constants c1 and c2 also posed a challenge in graphing. Recommendations for writing complex math equations included using LaTeX, with the options of bmatrix or pmatrix for matrices. The conversation also provided an example of using these options for a 2x2 matrix.
  • #1
Nathaniel Gossmann
2
0

Homework Statement


Pic_Ch9_1.jpg


I am working through problem #1, a-c.

Homework Equations


The main equations are dx/dt=Ax, (A-rI)v=0, and det(A-rI)=0.

The Attempt at a Solution



[/B]
Pic_Ch9.jpg

Here is my attempt. I am fairly confident in my answer to A. I'm less sure on my answer to B, however it is the same as the answer in the back of the book. My main problem is that I'm not sure how to start C. I understand how to do it with a single differential equation, however the matrices are throwing me off. I instinct says that I can simplify my equation in B to be a single 2x1 matrix. From there however, I am lost. Also, I'm not sure how the presence of the constants c1 and c2 would affect my graph. Any helpful pointers that lead me on the right path would be very helpful!
Thanks
 

Attachments

  • Pic_Ch9_1.jpg
    Pic_Ch9_1.jpg
    63.4 KB · Views: 735
  • Pic_Ch9.jpg
    Pic_Ch9.jpg
    57 KB · Views: 698
Last edited by a moderator:
Physics news on Phys.org
  • #2
We can't see the pictures you linked. Please try again. Use the UPLOAD button to put an image in a post.
 
  • #3
Yes, and please show your work directly, not as an image. We prefer that work be shown directly in the text pane, because work shown in images is difficult to read.
 
  • #4
anorlunda said:
We can't see the pictures you linked. Please try again. Use the UPLOAD button to put an image in a post.
Images are now fixed.
 
  • #5
Mark44 said:
Yes, and please show your work directly, not as an image. We prefer that work be shown directly in the text pane, because work shown in images is difficult to read.
What program do you recommend/is commonly used to write complex math equations?
 
  • #6
Nathaniel Gossmann said:
What program do you recommend/is commonly used to write complex math equations?
See our tutorial on LaTeX -- https://www.physicsforums.com/help/latexhelp/

Problem 1 looks like this:
##\frac {d \textbf x}{dt} = \begin{bmatrix} 3 & -2 \\ 2 & -2 \end{bmatrix} \textbf x##

My personal preference is for matrices to be in brackets. To surround them with parentheses, use pmatrix rather than bmatrix.

The unrendered script that I wrote looks like this: ##\frac {d\textbf x}{dt} = \begin{bmatrix} 3 & -2 \\ 2 & -2 \end{bmatrix} \textbf x##
 
  • #7
For part b, break out ##x_1(t)## and ##x_2(t)##, and show why their limits are as you say.
 
  • #8
Nathaniel Gossmann said:
What program do you recommend/is commonly used to write complex math equations?

Using "bmatrix" you get
$$\begin{bmatrix} 3 & -2 \\ 2 & -2 \end{bmatrix}$$.
Using "pmatrix" you get
$$\pmatrix{3 & -2 \\ 2 & -2} $$
Just right-click on each image and choose "show math as tex commands.." to see the syntax.
 
Last edited:

1. What is a system of differential equations?

A system of differential equations is a set of equations that describe the relationships between multiple variables and their rates of change over time. It is often used to model complex systems in physics, engineering, and other scientific fields.

2. What is the phase plane method?

The phase plane method is a graphical technique used to analyze and visualize a system of differential equations. It involves plotting the relationships between the variables on a two-dimensional graph, with each variable on a different axis.

3. What is the significance of the phase plane?

The phase plane allows us to see the behavior of a system of differential equations over time. By examining the shape and direction of the curves on the graph, we can determine the stability and possible equilibrium points of the system.

4. How is the phase plane used in real-world applications?

The phase plane method is widely used in a variety of fields, including physics, biology, and economics. It can help researchers understand and predict the behavior of complex systems, such as population dynamics, chemical reactions, and electrical circuits.

5. What are the limitations of the phase plane method?

While the phase plane method can provide valuable insights into the behavior of a system of differential equations, it is limited by the assumptions and simplifications made in the modeling process. It may not accurately capture all aspects of a real-world system and may require further analysis and refinement.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
388
  • Calculus and Beyond Homework Help
Replies
5
Views
834
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
454
  • Calculus and Beyond Homework Help
Replies
10
Views
973
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
130
  • Calculus and Beyond Homework Help
Replies
2
Views
520
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
489
Back
Top