If I was given three simultaneous linear ODEs, I can find the eigenvalues of the matrix. I'm assuming if I have a non-homogeneous system, I can just first disregard the non-homogeneous term and just compute the eigenvalues. If there also exist equilibrium points, how can I determine the stability of them just by eyeing the eigenvalues?
I know for a 2D case, two distinct negative eigenvalues give me a stable sink and so on. But what about the 3D case? How can I determine the stability of such a system? Do I really need to solve the entire problem and then figure out its long term behaviour? Also, how can I determine whether a quantity is increasing or decreasing just by eyeing this? Is this even possible?