Question

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

\({y^\prime } = x + y - 1\)

Short answer

The direction field for equation is figure labeled IV.

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\({y^\prime } = x + y - 1\)

To determine the direction field of the given differential equation first set\({y^\prime } = 0\), then, \(x + y - 1 = 0\)

From above equation, the right side of the equation is dependent on the independent variable\(x\)and dependent variable.

This means that the line segments along any horizontal or vertical lines are not parallel.

So, the direction field of the differential equation\({y^\prime } = x + y - 1\)must be given by figure IV.

i.e;