System of Differential Equations, Phase Plane

[Nathaniel Gossmann]

Homework Statement

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]

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

 

[anorlunda]

We can't see the pictures you linked. Please try again. Use the UPLOAD button to put an image in a post.

 

[Mark44]

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.

 

[Mark44]

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.

 

[Nathaniel Gossmann]

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?

 

[Mark44]

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##

 

[Mark44]

For part b, break out $x_1(t)$ and $x_2(t)$, and show why their limits are as you say.

 

[Ray Vickson]

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.