Partially decoupled linear system

342
0
1. The problem statement, all variables and given/known data

x' = x-3y
y'= -2y

There are two solutions of this system that lie on straight lines, in the sense
that the ratio y(t)/x(t) is constant along each such solution. Find these lines and the
corresponding solutions.

2. Relevant equations

None in particular that I can think of


3. The attempt at a solution

I don't follow the question. I see that it asks for 2 particular solutions that are straight lines but how does y(t)/x(t) come into this to help me solve for them?

thanks
 
31,927
3,893
1. The problem statement, all variables and given/known data

x' = x-3y
y'= -2y

There are two solutions of this system that lie on straight lines, in the sense
that the ratio y(t)/x(t) is constant along each such solution. Find these lines and the
corresponding solutions.

2. Relevant equations

None in particular that I can think of


3. The attempt at a solution

I don't follow the question. I see that it asks for 2 particular solutions that are straight lines but how does y(t)/x(t) come into this to help me solve for them?

thanks
I believe that where they're going with this problem is looking at the solutions if x' = 0 or y' = 0. Substitute 0 for x' and y' and you'll see that the solutions are straight lines, and the ratio y(t)/x(t) will make more sense.
 

HallsofIvy

Science Advisor
41,626
821
I'm afraid (and amazed) that Mark44 is mistaken here (or I am misunderstanding him- more likely that). You are NOT looking for "equilibrium solutions" (where x'= 0 and y'= 0). The only equilibrium solution is (0, 0). What you are looking for are straight line solutions through (0, 0).

If y= mx, then y'= mx'. Since y'= x- 3y and y'= -2y, that says that -2y= m(x- 3y). but y= mx so that is -2mx= m(x- 3mx)= m(1- 3)x. Since that is to be true for all x, -2m= m(1- 3m). Can you solve that equation for m? What lines does that give you?

(The values of m, by the way, will be eigenvalues of the coefficient matrix,
[tex]\begin{bmatrix}1 & -3 \\ 0 & -2\end{bmatrix}[/tex]
and the lines are in the direction of the eigenvectors.)
 
Last edited by a moderator:

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top