Determine limiting behavior as t goes to infinity(Differential Equation)

[Raziel2701]

Homework Statement

Determine the limiting behavior of solutions as t \rightarrow \infty, for all possible values y_0 = y(0)

Homework Equations

\frac{dy}{dt}=4-y^2

The Attempt at a Solution

I've obtained the constant solutions, they are -2 and 2. I sketched dy/dt to determine the monotonicity, and found that the constant solution 2 is stable. My question is whether this stable solution is what determines the behavior as t goes to infinity.

I just started taking differential equations and I'm trying to get used to the language and the nature of the questions. Since 2 is a stable solution, that should mean that all initial values of y should go to 2 as time passes correct?

 

[Dick]

No. A solution doesn't have to approach the stable solution unless it starts out near the stable solution. Otherwise, it might just go to infinity.