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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