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 point 2 is semi-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,

We have one critical point.

Find the phase portrait.

To find out where to point our arrows in the phase portrait, take for instance

And

From which we can see that both arrows aroundpoint up.

The phase portrait is shown below:

Since one arrow is towards toand one away from. Hence, the critical point is semi-stable.

Step 4:Sketch the typical solution.

We know that the solution curves will approach. If a curve begins belowtheline, it will approach it asgoes to infinity and if a curve starts above theline, it will approach it agoes to negative infinity.

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

Therefore, the critical point 2 is semi-stable.