This is sloppy working as it skips steps.
I'd do it like this...
I'm having difficulty understanding conventions for DE, and I suppose calculus in general. I have not had calculus in several years, so I'm trying to figure out what exactly is happening in this sequence.
Here's what the book says:
So, I understand the basics: rearrange the equation, integrate both sides, and solve for u(t).
My only concern is that I have no idea what the derivative notation means. For example, what exactly is happening from line 2 to line 3 that makes the du(t) drop out. Does this mean that when I integrate (du(t)/dt) (1/ u(t)) that du drops out? Ok, but why is the (d/dt) left? Doesn't that mean "take the derivative"?
^Is that what's happening? If so, then I still don't understand the next steps. Where is that d/dt going exactly??
I hope you can understand what I'm asking: what do all these d's, dt's, and d(u)'s mean. I usually just ignore them and remember the process.