How do slope fields work?

A slope fields for a differential equation of the form

#dy/dx=f(x,y)#

is a visual representation of the slope of a solution of the differential equation at each point. By observing a slope field, we can predict the behavior of the solution by starting with an initial point and following the flow created by the slope field.

Example

Let us look at the slope field for the differential equation

#{dy}/{dx}=y#,

which looks like:

Let us visualize a solution with the initial value #y(0)=2#; that is, we visualize a path following the flow of the slope field while passing through the point #(0,2)#.

The actual solution of the above initial value problem is #y=2e^x#, which looks like:

Did your prediction match the solution curve above? I hope so.

Let us try with this different initial value #y(0)=-1#. The solution is #y=-e^x#, which looks like:

As you can see in the examples above, you can easily visualize the solution curves of the differential equation even though we might not know the formulas for the solution.

I hope that this was helpful.