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 that instant and that in a period of $40$ years the population increased from $3,00,000$ to $4,00,000$.

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

where IN is (inflow concentration) $\times$ (inflow rate), and OUT is (current concentration $=x/V$) $\times$ (outflow rate), with $V$ the (possibly time-varying, but often constant when inflow rate = outflow rate) volume in the tank. This is a linear first-order equation in $x$, solved by the integrating-factor method; the initial amount of substance present fixes the constant of integration.

Common thread. In every one of these models, one physical reading (or a stated rate) determines the proportionality constant $k$ (or $C$), and a second condition either fixes the remaining constant or is the actual quantity asked for — so the general strategy is always: write the rate law, solve the resulting separable/linear equation for the general solution, then use the given numerical data to pin down the constants.