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Slope Fields











The technique of "Slope Fields" is a graphical technique used to solve differential equations. Slope Fields are most appreciated when an analytical solutions can't be found. There are several different ways to create the slope field and I will cover a few of them. First I will cover the "Point by Point" technique.

"Point by Point"

This technique is similar to when you first learned how to graph a function; that is, we start by making a table of values and then plot them on a graph.

For the differential equation y' = y+t, lets graph it on the interval from -2 to 2 for both the "t" and "y" axis.

Begin by creating a table.

You will pick different coordinates and then use those values for "y" and "t" to compute y' using the given differential equation.


Above the coordinates (0,0) were used to calculate y' = 0, and then a small line segment with a slope of zero was plotted on the graph at (0,0). The same procedure was also done for the coordinate (1,0). Remember that y' = m = slope.


The coordinates (0,1) and (1,1) were added to the table and their computed slopes were then plotted on the graph.

Try to complete the slope field on your own. When you are finished, your slope field should look similar to the picture below.


Now that the slope field is completed, let's see how it is useful. Given the differential equation y'=y+t and the initial condition y(0)=1/2, what is the long term behavior for y(t)? To answer this, start by plotting the initial condition on the graph of the slope field.

Then use the slopes on the graph as guides to determine the direction of the solution to y(t).

Looking at the graph we can see that this particular solution approaches infinity.


That's it!




After you feel comfortable with the above technique you can try using Isoclines to create the slope field.



Lee DeAnda

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