Q.Find the population of a city at any time $t$, given that the rate of increase of population is proportional to the population at the instant, and that in a period of 40 years the population increased from 30,000 to 40,000.

Let $P=P_0e^{kt}$, taking $t=0$ at the start of the given 40-year period, so $P_0=30{,}000$. At $t=40$, $P=40{,}000$: $40{,}000=30{,}000\,e^{40k}\Rightarrow e^{40k}=\dfrac43\Rightarrow k=\dfrac{1}{40}\log\dfrac43$. So the population at any time $t$ (years from that reference instant …

Model the situation as dx/dt proportional to x (or to (θ-θ0) for cooling), solve the resulting first-order linear/separable equation to get x = c·e^{kt} (or the Newton's-law form), then use the gi …

Mixing up which measured instant corresponds to t (measuring time from the WRONG reference point); forgetting that a ratio like x(t2)/x(t1)=e^{k(t2-t1)} lets k cancel out algebraically, and instead trying …