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I have a homework question in ODE and don't see limiting ratio mentioned anywhere in the notes. The question gives a two equation linear system solved by finding the eigenvalues and eigenvectors. It plots a few trajectories along with the eigenvectors and say to find the limiting ratio $\frac{y(t)}{x(t)}$.

Specifically it asks: You have enough information to be able to predict the limiting ratio $\frac{y(t)}{x(t)}$ as $t$ gets large for any trajectory that does not start on the line through ${(0,0)}$ determined by the sucking eigenvector.

Here are the equations:
$x'(t) = -0.26 x(t) + 0.9 y(t) $

$y'(t) = 0.07 x(t) + 0.06 y(t)$

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What have you tried doing? You presumably know how to solve that system, so you can try to actually compute the limit! –  Mariano Suárez-Alvarez Nov 19 '11 at 3:16
    
no but i have no clue at all what the limiting ratio means. –  riotburn Nov 19 '11 at 3:22
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If $x(t)$ and $y(t)$ are the solutions, then the limiting ratio is $$\lim_{t\to\infty}\frac{y(t)}{x(t)}$$ So, you are right, they omitted the mention of $t\to\infty$. –  GEdgar Nov 19 '11 at 3:33
    
Oh, so what is y(t)/x(t) then? –  riotburn Nov 19 '11 at 3:39
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Didn't GEdgar tell you a few lines up what the limiting ratio is? –  Gerry Myerson Nov 19 '11 at 11:21
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1 Answer

So I figured it out:

If you want to find the limiting ratio of a 2 linear equation diffeq, you need to take the slope of the eigenvector corresponding to a positive eigenvalue. By slope I mean, if your eigenvector is {1.5, 3}, then you would plot a line that goes through {1.5, 3} and its opposite {-1.5, -3} and find the slope from there.

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