I have a simple example from my physics text using a separable ODE, but I don't understand the steps that they took to solve it. It should be pretty straightforward, but I am getting something different when I try to solve. I've attached the image of the page where my questions are, and a red "1" and "2" indicates my problems with it.
Note: C = capacitor, E = emf of battery, q = charge in circuit
#1: I don't understand how they got the final answer here. If you multiply (CE - q)/RC by -1, you'd get (q-CE)/RC. So how did they get -(q-CE)/RC ? Seems like there's an extra negative sign.
#2: I don't understand the integral solved on the left side of int((dq/(q-CE)) to get what they have for ln((q-CE)/(-C)).
I would have thought that int((dq/(q-CE)) = ln(q-CE)
I know these seem simple, but an I missing something obvious here in these?
In red 1, CE-q numerator, they just pulled out the "-" and made it -(-CE+q) = -(CE-q).
In #2, same type of arithmetic, just manipulate the equation without changing its quantity; remember you can do the same thing to each side of an equation and it doesn't effect the result.
Hope this helps.