Question

A critical point c of an autonomous first-order DE is said to beisolated if there exists some open interval that contains c but noother critical point. Can there exist an autonomous DE of the form given in (2) for which every critical point is nonisolated?Discuss; do not think profound thoughts.

Short answer

Yes, all critical points are either unstable or semi-stable.

Step-by-step solution

Step 1:Critical points

The zeros of the function f in $dy/dx=f\left(y\right)$are of special importance. Wesay that a real number c is a critical point of the autonomous differential equation $dy/dx=f\left(y\right)$ if it is a zero of f—that is, $f\left(c\right)=0$. A critical point is also called an equilibrium point or stationary point.

Find the value of critical points

The differential equation,$\frac{dy}{dx}=f\left(y\right)$

Yes every critical point can be non-isolated if there are no points on the phase plot that are asymptotically stable.

In other words all critical points are either unstable or semi-stable.

Therefore, all critical points are either unstable or semi-stable.