Question

In Problems 29 and 30 consider the autonomous differential equation dy/dx = f(y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each differential equation. By hand, sketch typical solution curves in the sub-regions in the xy-plane determined by the graphs of the equilibrium solutions.

Short answer

Critical points are 0 and c.

Step-by-step solution

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.

Critical points and phase portrait

Critical points are the solutions of $f\left(y\right)=0$.

From the graph, it is clear that the solutions are 0 and c.

We have critical points $y=0,y=c$.

From the graph we can also see that before 0, f is positive, between 0 and c the graph is negative and after c the graph is positive, from which we can see that arrows point up, down, up. The phase portrait is shown below:

Since both arrows are pointing away from c, it is an unstable critical point. Since both arrows are pointing toward 0, it is a stable critical point.

Solution curve

Some typical solution curves look like this,

Therefore, Critical points are 0 and c.