Question

How do slope fields work?

Answer 1

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.

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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.

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I hope that this was helpful.