I covered differential equations a long time ago and so was confused when my Population Ecology textbook displayed the following "proof" to solve $\frac{dN}{dt} = rN$ for $N$.
My (probably stupid) question is, they say they take the integral from 0 to $t$ of both sides, but why is the integral on the left-hand side from $N(0)$ to $N(t)$? I understand the rest of the derivation just fine.
Thank you!
Doing it step by step you get the equation to be equivalent to $\frac{dN(s)}{ds} \cdot \frac 1N = r$ (where I changed the variable $t$ to $s$). Then we integrate both sides with respect to $s$ from $0$ to $t$ (I don't want to integrate with respect to $t$ if the upper bound is also called $t$).
We get: $$r\int_0^t ds = \int_0^t \frac{dN(s)}{ds} \cdot \frac 1N \ ds = \int_{N(0)}^{N(t)} \frac 1N \ dN(s)$$ with the substitution formula: new variable is $N(s)$ and then $dN(s) = \frac{dN(s)}{ds} ds$ which cancels with the first part of the second integral.