Question

In Problems, 21–28 find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in theplane determined by the graphs of the equilibrium solutions.

Short answer

Answer:

The critical pointsandare stable and the critical point is stable.

Step-by-step solution

Critical points.

The zeros of the functioninare of special importance. Wesay that a real numberis a critical point of the autonomous differential equationif it is a zero ofthat is,. A critical point is also called an equilibrium point or stationary point.

Find the critical point and phase portrait.

To find the critical points, equate.

So,

Hence, the critical points of the given differential equations areand.

The phase portrait is shown below:

Since both arrows are pointing away fromand, so they are unstable critical points.

Since both arrows are pointing towards, it is a stable critical point.

Sketch the typical solution.

A sketch of the family of typical solution curves of the given differential equation is shown below:

Therefore, the critical pointsandare stable and the critical pointis stable.