# Partially decoupled linear system

#### hover

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

#### Mark44

Mentor
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

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,
$$\begin{bmatrix}1 & -3 \\ 0 & -2\end{bmatrix}$$
and the lines are in the direction of the eigenvectors.)

Last edited by a moderator:

#### Mark44

Mentor
I'm afraid (and amazed) that Mark44 is mistaken here (or I am misunderstanding him- more likely that).
It can happen that I am mistaken.

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