Qualitative Techniques: Slope Fields 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
:
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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:
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
.
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.
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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.
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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!
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Let's consider the following differential equation:
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.
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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.
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:
Note that this equation is also autonomous!
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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
Now the slope field looks slightly more complicated.
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Here is the same slope field again. What is special about the points on the red parabola?
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Author: Helmut Knaust