Question

The graph of the function f(x) is, and the the horizontal axis is x.

Given the differential equation x?(t)=f(x(t))x?(t)=f(x(t)).
List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equations are stable, semi-stable, or unstable. (The second line after the comma has a drop menu that asks for stable, semi-stable, or unstable.)

A.) ________ , ________

B.) ________ , ________

C.) ________ , ________

D.) ________ , ________

Answer 1

The equilibrium points are the points where the curve touches or crosses the horizontal axis.

So, the equilibrium points are x=-4, -1, 1 and 5/2

Now, f(x) is negative in (-inf, -4) and positive in (-4, -1) so, x=-4 is unstable.

f(x) is positive in (-4, -1) and positive in (-1, 1) so, x=-1 is semi-stable.

f(x) is positive in (-1, 1) and negative in (1, 5/2), so x=1 is stable.

f(x) is negative in (1, 5/2) and positive in (5/2, inf), so, x=5/2 is unstable.