Chapter 2.8, Problem 1E

Interpretation Introduction

Interpretation:

The slope is constant along the horizontal lines for the given slope field is to be proved.

Concept Introduction:

The slope of the curve is a derivative at a point on the curve.

$\dot{\text{x}}$ completely depends only on parameter x and independent of parameter t

Answer to Problem 1E

Solution:

The slope is constant along the horizontal lines for the given slope field as proved below.

Explanation of Solution

From the given differential equation $\dot{\text{x}}\text{=x(1-x)}$, the variable $\dot{\text{x}}$ explicitly depends on x and completely independent of t. For any horizontal line of the slope field, x remains constant for any value of t. Hence, the slope is constant along the horizontal lines.

Conclusion

For a slope field, $\dot{\text{x}}$ explicitly depends on x and completely independent of t.