Date: Feb 10, 2010 11:39 AM
Author: Skerbie
Subject: [ap-calculus] More on integral of 1/x and initial conditions

Greetings all,

I was searching the archives looking for
discussions regarding the antiderivative of 1/x, resulting in ln|x| vs.
ln(x). (I know Lou Talman has a nice write-up on this one ...) At one
point, Jon Rogawski posted the following:

*****
<snip from April 9, 2008>

A similar issue arises in a much more common setting. Consider the
standard formula

integral of 1/x dx = ln |x| + C

This
tells us that y = ln |x| + C is the general antiderivative of 1/x. In
other words, it's the general solution of y' = 1/x. But this is not
completely correct. If we were so inclined, we could choose different
constants for x>0 and x<0. For example,

y = ln x for x > 0
y = ln |x| + 10 for x < 0

is a valid (but rather unnatural!) antiderivative of 1/x.

*****

I'm
fine with all this theoretically. My question is this ... would this
occur in "real life"? In other words, are there practical situations
in which practically-minded models have solutions that occur across
both sides of a discontinuity and hence require differing choices for
the value of the constant of integration? I don't think a practical
situation would have two differing initial conditions simultaneously,
but could you give examples of cases from practical models in which the
same "solution" (piecewise or otherwise) could have initial conditions
that occur on either side of the discontinuity? I've seen plenty of
modeling situations in which we can have different IC's on either side
of equilibria within the same context, but that's not quite the same
thing.

Or is it the case that in *most* practical situations, the most "typical" solution curves of interest would all be on a particular "side" of the discontinuity?

Does my question make sense?

Thanks,
Bill




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