Hi,
In class today, we worked out the following question. However, I don't understand why the teacher said we can't differentiate ydx because it will be equal to zero, as well as where the 'd' disappeared to (please see my comments in red below)
Question: Solve for the following differential equation for the original equation:
(you cannot differentiate ydx, ydx = 0)
(where did the d go? Why not )?
To separate the variables, you need the terms to be with the and the terms to be with the .
So this should really be (if you're going to use the lazy notation...)
...
Personally, I prefer using the "reverse chain rule" approach (you end up with the same integral but is more mathematically correct, since it doesn't treat as a fraction)...
, which gives us the same integrals.
Go from here...
Not that it matters in the end (let's not be dogmatic)... but, you might seem, at least, to be cancelling a fraction when you do this:
One way to 'correct' such an impression is of course by replacing d with w.r.t., ... or else... ooh, I don't know, how about a diagram -
, which gives us the same integrals.
i.e.
... where (key in spoiler)
Spoiler:
What's topsquark doing then? If I'm reading it right, using the fraction notation to get us up onto the integral level via the product rule, which in this case at least avoids having to go in and out of logs. Which is useful. So, re-writing this
as
we can present his version like this...
i.e.
... where
Spoiler:
I have to admit that all that didn't enable me to decide whether Sparky's teacher was doing either of these things or something else.
Cannot integrate? Or were they referring to an integration by parts / product rule type maneouvre? Could be, because the next line is where topsquark and the second pair of diagrams (above) arrived at. Apart from
I think they just left an integral sign in there by mistake, after integrating.(where did the d go?)?
I don't follow. Are you integrating with respect to x, perhaps?(Why not )?
Anyway, I hope this helps, or is of interest.
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