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April 4th, 2019, 10:27 AM   #1
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Steady States

For an example problem to solve in an engineering class I was given the following systems and told to find the steady states of each and the eigenvalues for each steady state.

1)
$\displaystyle dx/dt = x(1-2y-x)$
$\displaystyle dy/dt = y-x$

2)
$\displaystyle x' = x(10-x-y)$
$\displaystyle y' = y(30-2x-y)$

I know that for steady states I need to solve for when the derivatives are both equal to zero. For both systems I know that x=y=0 is a trivial solution. I found another solution to be x=y=1/3 for System #1 and x=20, y=-10 for System #2.

My problem is, I remember how to find eigenvalues of matrices from Lin. Alg. but I'm not sure how to get these equations into a form where I can solve for the eigenvalues. How would I go about finding the eigenvalues for these systems?
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