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 pointis unstable, the critical pointis semi-stable and the critical pointis asymptotically 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.

To find the critical points, equate.

So,

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

Now we put the critical points on a vertical line and divide the line into four intervals defined by

Find the phase portrait.

The phase portrait is shown below:

The arrows are moving away from the critical points. Hence, the critical pointis unstable.

At the critical point, the curve above it moves away from it, while arrows below it move towards it. Hence, the critical pointis semi-stable.

The arrows are pointing towards the critical point. Hence, the critical pointis asymptotically stable.

Sketch the typical solution.

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

Therefore, the critical pointis unstable, the critical pointis semi-stable and the critical pointis asymptotically stable.