Problem

Match the following differential equations to their slope fields.

1.

2.

3.

A.

B.

C.

Solution

There are a number of ways to check if a particular differential equation matches a given slope field. One could solve the differential equation and graph particular solutions, and compare them against the slope field, for example.

However, a method that can be much easier is to choose a set of points (x, y) that are particularly easy to plug into the differential equation, and compare the result with the slope of the minitangent lines in the slope field around the points (x, y).

For example, the differential equation 1 equals zero whenever x = 0. Therefore, the slope field associated with 1 will have horizontal lines along the line x = 0. Only slope field B has this property.

Another way to identify which slope field matches which equation is to notice what appears in the slope field when the differential equation involves only one variable. In differential equation 2, only x appears on the right hand side. Therefore, we should look for a slope field whose lines do not change when y changes. Slope field C has this property.

Finally, the differential equation in 3 is zero whenever y = 2; only slope field A has lines with zero slope along y = 2.