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. 😀

Overall, solutions to these differential equations are functions or sets of functions that satisfy the conditions specified by the equation and provide insight into the underlying processes being modeled.

In many cases, a single differential equation may have multiple solutions, which are referred to as families of functions. These families of functions may differ in their behavior, shape, and mathematical properties, but they all satisfy the equation and provide a description of the system being modeled. For example, in physics, the solutions to a differential equation describing the motion of a particle may include a variety of functions that describe the particle's position and velocity over time.

The concept of solutions to differential equations is fundamental to the study of dynamic systems and plays a critical role in the analysis and prediction of complex phenomena. By finding the solutions to a differential equation, researchers and engineers can gain insight into the behavior of a system and make predictions about its future behavior. This information is used in various applications, including the design of engineering systems, the prediction of financial trends, and the modeling of physical and biological systems.

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.