Question

Let y(????)y(t) be a solution of y˙=17y(1−y7)y˙=17y(1−y7) such that y(0)=14y(0)=14. Determine lim????→[infinity]y(????)limt→[infinity]y(t) without finding y(????)y(t) explicitly.

Answer 1

Explanation:

Given that,

y' = 17y ( 1-y^7)

Let y=1

Then, y' = 0 for all t

Then show that it is the only stable equilibrium point so that as y→1, t→∞ with any initial value.

So, the graph solution will be

y(0) = 1 and this will be an horizontal line

If, y(0) > 1 then, y' < 0 by inspecting the first equation, so the graph is has decreasing solution.

Likewise, if y(0) < 1 then, y' > 0 and the graph is increasing.

So no matter the initial condition, graph of the solution will be asymptotic to the horizontal line above.

This make the limit be 1.

This shows that x = 1 is a stable equilibrium.