Chapter 7.2, Problem 16E

To determine

To draw : the direction field using computer algebra system and sketch on it the solution curve that passes through the point.

Explanation of Solution

Given information :

The differential equation is $y\text{'}=x\left(y^2-4\right)$ .

Graph :

By using computer algebra system,

The direction field is obtained as:

  

To sketch the solution curves first sketch the solution, for sketching solution start at the origin and move to the right in the direction of the line segment (which has slope 1), then continue the solution curve so that it moves parallel to the nearby line segments. And for more solution curve change the y -intercept.

Since solution graph passes through the point $\left(0,1\right)$ , so the slope intercept for that curve,

  $\begin{array}{l}y\text{'}=x\left(y^2-4\right)\\ y\text{'}=0\left(1-4\right)\text{ [substitute }x\text{ by 0 and }y\text{ by 1]}\\ y\text{'}=0\text{ [simplify]}\end{array}$

So the graph can be observed as:

  

Using implicit plotting capability of a CAS to graph the curve

The graph can be obtained as :

  

Interpretation : from the above graph it can be observed that the graph by direction field and CAS are same.