I'm reading an explanation of how to solve first-order differential equations. Part of the way through, I have this:
If $\frac{dR}{dx}=RP$, $\begin{align*} \frac{dR}{R} &= Pdx, \\ \int \frac{dR}{R} &= \int Pdx, \\ \ln R &=\int Pdx +c. \end{align*} $
Now, this makes me uneasy: I don't like to separate variables. I do it when I integrate with a substitution, but I'm perfectly aware when I'm doing it that I've slipped out of math for a second to manipulate my symbols for convenience, and I could do it more formally if I had to.
That's what I want to do here, but I can't figure it out. Given the first line, how do I solve for R without separation of the variables?
I should add that P and R are both functions of x.