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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) $

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

  • 2
    What have you tried doing? You presumably know how to solve that system, so you can try to actually compute the limit!2011-11-19
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    no but i have no clue at all what the limiting ratio means.2011-11-19
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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 $t\to\infty$.2011-11-19
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    Oh, so what is y(t)/x(t) then?2011-11-19
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    @riotburn You would have to solve the system given. The solutions would be $y(t)$ and $x(t)$. There are many ways to solve that system, but as it stands your question was asking about the limiting ratio. Do you also need to know how to solve the system?2011-11-19
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    No I know how to solve the system, just dont know what the limiting ratio is and how to solve it. Am i just dividing the solutions i get for y(t) and x(t) and then taking the limit?2011-11-19
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    Didn't GEdgar tell you a few lines up what the limiting ratio is?2011-11-19
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    Yes, but I'm just confirming that that is what I am supposed to do...take the solutions, divide and take the limit.2011-11-19

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