How do slope fields work?

Question

Elijah Abram · Accepted Answer

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.

Adam Adkins · Answer

undefined Slope fields, also known as direction fields, are graphical representations used to visualize solutions to ordinary differential equations (ODEs). They work by illustrating the slope of the solution curve at various points in the xy-plane.

To create a slope field for a first-order ODE of the form $ \frac{dy}{dx} = f(x, y) $, where $ f(x, y) $ is a given function, follow these steps:

1. Choose a set of $ x $- and $ y $-values that cover the region of interest.
2. At each point $ (x, y) $, calculate the slope $ f(x, y) $.
3. Draw a short line segment at each point with the calculated slope. These line segments represent the direction in which the solution curve would move if passing through that point.
4. Repeat this process for enough points to create a grid of line segments covering the region.

By examining the slope field, you can gain insight into the behavior of solutions to the differential equation. Solution curves will tend to follow the direction indicated by the line segments, providing a qualitative understanding of the behavior of the system.

Slope fields are particularly useful when exact solutions to ODEs are difficult or impossible to find analytically. They provide a visual aid for understanding the behavior of solutions without the need for complex calculations.