Qualitative Techniques: Slope Fields

A differentiable function--and the solutions to differential equations better be differentiable--has tangent lines at every point. Let's draw small pieces of some of these tangent lines of the function tex2html_wrap_inline32 :

Slope fields (also called vector fields or direction fields) are a tool to graphically obtain the solutions to a first order differential equation. Consider the following example:

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The slope, y'(x), of the solutions y(x), is determined once we know the values for x and y , e.g., if x=1 and y=-1, then the slope of the solution y(x) passing through the point (1,-1) will be tex2html_wrap_inline50 . If we graph y(x) in the x-y plane, it will have slope 2, given x=1 and y=-1. We indicate this graphically by inserting a small line segment at the point (1,-1) of slope 2.

Thus, the solution of the differential equation with the initial condition y(1)=-1 will look similar to this line segment as long as we stay close to x=-1.

Of course, doing this at just one point does not give much information about the solutions. We want to do this simultaneously at many points in the x-y plane.

We can get an idea as to the form of the differential equation's solutions by " connecting the dots." So far, we have graphed little pieces of the tangent lines of our solutions. The " true" solutions should not differ very much from those tangent line pieces!

Let's consider the following differential equation:

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Here, the right-hand side of the differential equation depends only on the dependent variable y, not on the independent variable x. Such a differential equation is called autonomous. Autonomous differential equations are always separable.

Autonomous differential equations have a very special property; their slope fields are horizontal-shift-invariant, i.e. along a horizontal line the slope does not vary.

Try it yourself!

What is special about the solutions to an autonomous differential equation?

Here is an example of the logistic equation which describes growth with a natural population ceiling:

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Note that this equation is also autonomous!

The solutions of this logistic equation have the following form:

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As a last example, we consider the non-autonomous differential equation

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Now the slope field looks slightly more complicated.

Here is the same slope field again. What is special about the points on the red parabola?

If you would like more practice, click on Example.

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Author: Helmut Knaust

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