Problem

Match the following differential equations with their slope fields.

1.

2.

3.

4.

A.

B.

C.

D.

Solution

We match the slope fields to the equations by looking for distinguishing characteristics about the slope field. For example, in slope field A, the minitangent lines are horizontal (slope is 0) when x = –1 or y = 0. Therefore the differential equation that matches with A must equal 0 when x = –1 or y = 0; this equation must be 4.

Similarly, the lines in slope field B are horizontal when y = 0 or y = 1; only the differential equation 2 is zero when y = 0 or y = 1. Therefore B matches with 2.

The lines in slope field C are sloped positively for y > 0 and negatively for y < 0. This is consistent with equation 3, where the factor of sin(y) has similar positive and negative characteristics.

Finally, the lines in slope field D are horizontal along the line y = x. The differential equation in 1 is zero whenever y = x, so D matches with 1.