Given the differential equation x′=−(x+3)(x+1)3(x−1)2(x−2.5). list the constant (i. e. equilibrium) solutions to this differential equation in increasing order and indicate whether or not these solutions are stable, semi-stable, or unstable.

=-3       Stable

=-1       Instable

=1        Semistable

=2.5    Instable

Step-by-step explanation:

Equilibrium constant solutions (or critical points) occur whenever x′ = f(x) = 0

x′=−(x+3)(x+1)3(x−1)2(x−2.5)=0

=-3

=-1

=1

=2.5

we can see the plot of x′ = f(x) in the figure annexed. In order to analyse the stability of the constant solutions, we must see how the function changes around the constant solutions X's:

(i.) If f(x) < 0 on the left of X, and f(y) > 0 on the right of

X, then the equilibrium solution is unstable.

(ii.) If f(X) > 0 on the left of X, and f(y) < 0 on the right of

X, then the equilibrium solution y = c is stable.

(iii.) If f(x) > 0 on both sides of X, or f(x) < 0 on both

sides of c, then the equilibrium solution y = X is

semistable.

Then:

=-3       Stable

=-1       Instable

=1        Semistable

=2.5    Instable

17.08

correct me if i am wrong.