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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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    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

1 Answers 1

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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.