Question

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.

Answer 1

`X_(1)`=-3 Stable

`X_(2)`=-1 Instable

`X_(3)`=1 Semistable

`X_(4)`=2.5 Instable

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

`X_(1)`=-3

`X_(2)`=-1

`X_(3)`=1

`X_(4)`=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:

`X_(1)`=-3 Stable

`X_(2)`=-1 Instable

`X_(3)`=1 Semistable

`X_(4)`=2.5 Instable