Chapter 7.2, Problem 17E

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 $y\text{'}=y^3-4y$ .

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 $\left(0,c\right)$ , so the slope intercept for that curve,

  $\begin{array}{l}y\text{'}=y^3-4y\\ y\text{'}=c^3-4c\text{ [substitute }x\text{ by 0 and }y\text{ by 1]}\\ y\text{'}=c\left(c^2-4\right)\text{ [simplify]}\end{array}$

For various value of c solution will be different,

So the graph can be observed as:

  

To find the value where the limit exist,

Because in the solution of the differential equation, log does not take negative and zero value, so limit will be exist if the value of $c>2$ .

The value of limit is always $-2,0,2$ because limit exist only for $c>2$ .

Interpretation : from the above graph it can be observed that Because in the solution of the differential equation, log does not take negative and zero value, so limit will be exist if the value of $c>2$ .

The value of limit is always $-2,0,2$ because limit exist only for $c>2$ .