In my “business calculus class”, we were studying implicit differentiation.
We had a problem:
I showed them three ways to do the problem, all of which yield different looking answers:
Differentiate both sides: which yields:
Method 2: do directly:
This leads to
Of course that looks different; but we can always solve for and do it directly:
which yields which yields the easy solution:
Now one can check that all three solutions are in fact equal on the domain
But here is the question that came to mind: in the first two methods we had two different two variable equations:
So what does this mean for ? Is it uniquely determined?
of course it is: what we really have is whose solution IS uniquely determined on an open rectangle so long as and are continuous and and are continuous also. 🙂
But I didn’t realize that until I took my morning swim. 🙂
This is the value of talking to a friend who knows what he/she is doing: I was reminded that means that and indeed have unique solutions that have the same slope at a common point, but with just this there is no reason that the solutions coincide over a whole interval (at least without some other condition).
So we have something to think about and to explore; I don’t like being wrong but I love having stuff to think about!
Now, of course, we have “different” differential equations with the same solution; yes, there is a theory for that. I’ve got some reading to do!