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Doug Kuhlmann

Posts: 3,630
Registered: 12/6/04
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Posted: Feb 24, 1998 9:44 AM
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I have always assumed that slope fields (direction or gradient fields) were
relatively easy, and still do. Yesterday while working with a colleague I
came across what could be a confusing point. Consider the function y=e^x.
This function satifies the differential equation y'=y (1) and also
satisfies the differential equation y'=e^x (2). However, if you sketch the
slope field for each of these two diff eqs you get very different slope
fields. The first, since y' is a function of y alone, has constant slopes
along horizontal lines. The second, since y' is a function of x alone, has
constant slopes along vertical lines.

The family of curves that satisfiy (1) is y=Ae^x, where A is an arbitrary
constant. The family that satisfiy (2) is y=e^x+C where C is arbitrary.
It turns out that y=e^x is in both families.

The confusion arose when a fellow tired to reverse the usual problem: Given
a function (in this case y=e^x) find the slope field that this function
"lives in". The answer is not unique.

This might not be a major point, but it is the first time I've seen some
confusion with this.


Doug Kuhlmann
Math Department Chair
Phillips Academy
180 Main Street
Andover, MA 01810

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