7.4 Reasoning Using Slope Fields

Continued from 7.3 Sketching Slope Fields.

The actual solution (which can actually be manipulated to be separable*) to the differential equation in Eq. 39 is the following:

By going to https://www.desmos.com/calculator/fjli4efhcj and clicking the play button on equation 18, you can see that this curve does indeed fit the given slope field for any constant C. Clicking the play button will show different curves for different values of C.

The most intuitive way to think of a slope field is to picture a fluid flowing and then placing an object on the fluid that will trace out a path. This path is approximate to the solution to the curve that represents the differential equation. 😀

Fill in the table below for different values of y’ at different coordinate points. Use a calculator to find the values to two decimal places. Create a slope field and then solve the differential equation and confirm that your slope field matches the solution to the differential equation. ✍

Varying values of C plotted over the slope field are shown here:

*One can make this separable by doing a substitution:

From here, one can solve the latter differential equation (which will give a solution that is only a function of x) and substitute this into u = x + y, which will give the aforementioned solution to the original differential equation.