Question

\({\bf{21}} - {\bf{24}}\)Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer.

\({y^\prime } = x(2 - y)\)

Short answer

The direction field for equation is figure labeled I.

Step-by-step solution

Definition

The direction field is defined as the collection of small line segments passing through various points having a slope that will satisfy the given differential equation at that point.

Explanation

The given differential equation \({y^\prime } = x(2 - y)\)

Put\({y^\prime } = 0\), solve as follows:

\(x(2 - y) = 0\)

This implies\(y = 2\)

Thus, slope of the direction field of the differential equation \({y^\prime } = x(2 - y)\)is horizontal

So, the match is either I or III.

Since the equation right side of the equation is dependent on the independent variable\(x\) this implies that the line segments along any horizontal line are not parallel.

So, the direction field of \({y^\prime } = x(2 - y)\)must be the figure labeled I.

i.e;