Implicit Differentiation and Differential Equations

In my “business calculus class”, we were studying implicit differentiation.
We had a problem:
Find if
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?

Answer:  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!