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?