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Posted by Subhotosh Khan on September 02, 2002 at 10:54:51:

In Reply to: DE equilibrium points -- am I on the right track? posted by colin on August 30, 2002 at 13:37:10:

: We're given the eq. dy/dt = f(y)
: suppose that y1 is a critical point: f(y1) = 0

: We're asked to show that the constant equilibrium solution g(t) = y1 is:
: asymptotically stable if f'(y1) < 0
: asymptotically unstable if f'(y1) > 0

: ------------------------------

: I think I know how to solve this but need a sanity check...

: f'(y1) is the derivative of f(t).
: if f'(y1) is negative, then it means the slope of f(t) is downward around the point y1, and since y1 is a zero of f(y) then f(y) must be positive when y < y1 and negative when y > y1. Thus it is asymptotically stable at y1.
: if f'(y1) is > 0, then it's just the opposite and asymp. unstable.

: is this a satisfactory & complete answer?
: or is there some more rigorous formal way to prove this?

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