Question

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

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

Short answer

The direction field is III.

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

Put\({y^\prime } = 0\), we get

\(\begin{aligned}{{}{}}{2 - y = 0}\\{y = 2}\end{aligned}\)

So, the slope of direction field of differential equation at\(y = 2\)is zero by definition of direction field.

The possible direction fields are I or III.

Since the independent variable\(x\)does not occur on the right side of the equation\({y^\prime } = 2 - y\),so the line segments along any horizontal line are parallel.

Thus, the direction field of\({y^\prime } = 2 - y\)is the figure labeled III.

i.e;