I am watching this video where the guy is deriving the equations for the step function of a series RC circuit.
At some point he has the following equation:
\$ \frac{dt}{RC} = \frac{dv}{v_s – v} \$
and he suddenly multiplies both sides by -1 in order to get a minus sign on the left side and invert \$v \$ and \$v_s \$
\$ -\frac{dt}{RC} = \frac{dv}{v – v_s} \$
He justifies that by saying "lets do that because we want to write v minus v_s"
what?
Obviously he knows that that minus sign will be the exponential exponent, making the final result correct but this is not a mathematical justification.
Do you guys have any idea why?
NOTE: you can see the sign change on the right side, on the line the integral symbol appears for the first time.
Best Answer
It's the same either way. You don't have to multiple by -1. Try it:
$$\begin{align} \frac{dt}{RC} &=\frac{dv}{v_s-v}\\ \\ \int_{v_0}^v \frac{dv}{v_s-v} &= \int_{0}^t \frac{dt}{RC}\\ \end{align}$$
let's forget about the right side, since it doesn't pertain to your question.
Using
$$\begin{align} \int \frac{1}{ax+b}dx = \frac{1}{a}ln|ax+b|\\ \end{align}$$
then \$a=-1\$ and \$b=v_s\$ so
$$\begin{align} \int_{v_0}^v \frac{dv}{v_s-v} \rightarrow(-1)ln|v_s-v|\\ \end{align}$$
now follow the rest of the tutorial and you'll eventually end up with
$$\begin{align} -v &= -v_s + (v_s-v_0)e^{\frac{-t}{RC}}\\ \end{align}$$
at which point, obviously, to solve for \$v\$, multiply both sides by -1.