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)
[MATH]dx/dt = x(1-2y-x)[/MATH][MATH]dy/dt = y-x[/MATH]
2)
[MATH]x' = x(10-x-y)[/MATH][MATH]y' = y(30-2x-y)[/MATH]
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?
1)
[MATH]dx/dt = x(1-2y-x)[/MATH][MATH]dy/dt = y-x[/MATH]
2)
[MATH]x' = x(10-x-y)[/MATH][MATH]y' = y(30-2x-y)[/MATH]
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?