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)