For any differential equation of the form dy/dx = f(x,y), we can construct a “slope field” for the differential equation, which allows us to understand the qualitative behavior of the solutions without needing to know a formula for the solutions. The following set of notes introduces the main ideas, and has some exercises for you to try for practice.
There are many free online applets that generate slope fields, but this one from Bluffton University is particularly good. You can click on the slope field to see the graph of the solution that goes through the point you clicked.
This Desmos graph gives you another way of generating the slope field for any differential equation dy/dx = f(x,y). Before trying different expressions in x and/or y, make sure you understand how the slope field depends on the constant k and agrees with what we already know in the case of the exponential growth and decay equations y’ = ky.