Exponential Models with Differential Equations AB & BC

Every proportional-rate model solves to $y = y_{0}e^{kt}$, and the constant carries the behaviour: positive $k$ grows, negative $k$ decays, and the units of $k$ are one over time, so it is neither an amount per unit time nor a duration. Doubling time and half-life come from setting the final value to a multiple of the initial one, which cancels $y_{0}$ and leaves $t = \frac{\ln 2}{k}$ in size; a half-life of $5730$ years gives $k \approx -1.21 \times 10^{-4}$ per year, and after $n$ half-lives the fraction left is $\left(\frac{1}{2}\right)^{n}$.

Newton's law of cooling, $\frac{dT}{dt} = k(T - T_{a})$, produces a SHIFTED exponential: the difference from the surroundings decays, not the temperature itself, so $T = T_{a} + Ce^{kt}$ with $C = T_{0} - T_{a}$. Dividing by $T - T_{a}$ to separate discards the constant solution $T = T_{a}$, which this family happens to recover at $C = 0$. The same shape covers any quantity approaching a limit $L$: $y = L + Ce^{kt}$, with $y = L$ as its equilibrium.