7.3 Sketching Slope Fields

Slope fields allow us to visualize a solution to a differential equation without actually solving the differential equation. Let’s construct a slope field to solidify this idea. 🧠

Slope fields essentially draw the slopes of line segments that go through certain points. Let’s consider the following differential equation:

The slope (m) at point (x, y), in this case, is just x + y, which we can put into a table for various coordinates:

We can use this data to draw an approximate solution to the differential equation by drawing short line segments through each point that have the corresponding slope: 🌄