Q.The rate at which the population of a city increases at any time is proportional to the population at that time. If there were 1,30,000 people in the city in 1960 and 1,60,000 in 1990, what approximate population may be anticipated in 2020? $\left[\log_e\left(\dfrac{16}{13}\right) = 0.2070,\ e^{0.42} = 1.52\right]$

A single class of first-order differential equation models a wide range of real phenomena: whenever a quantity's instantaneous rate of change depends on the quantity itself (and possibly time), the governing equation has the form $\dfrac{dx}{dt}=f(x,t)$.

Population growth (Malthusian law). If a population $x(t)$ grows at a rate proportional to its current size, $\dfrac{dx}{dt}=kx$ with $k>0$. This is separable: $\dfrac{dx}{x}=k\,dt\Rightarrow\ln x=kt+C\Rightarrow x=x_0e^{kt}$, where $x_0$ is the population at $t=0$. Doubling/tripling-time problems fix $k$ from one data point and then answer a second question using the same $k$.

Radioactive decay. The same law with a negative rate constant, $\dfrac{dA}{dt}=-kA\ (k>0)$, models decay: $A(t)=A_0e^{-kt}$. The half-life is the time $t_{1/2}$ at which $A=\dfrac{A_0}{2}$, found by solving $\tfrac12=e^{-kt_{1/2}}$.

Newton's law of cooling/warming. The rate at which a body's temperature $T(t)$ changes is proportional to the difference between $T$ and the constant ambient temperature $T_m$:

Separating and integrating gives $T-T_m=Ce^{kt}$; the constant $C$ is fixed by the temperature at $t=0$, and $k$ is fixed from one later reading, after which the model predicts the temperature (or the time to reach a target temperature) at any other instant.

Mixture (tank) problems. For a substance flowing into and out of a well-stirred tank, the amount $x(t)$ present obeys

$\dfrac{dx}{dt}=\text{IN}-\text{OUT},$ …