Question

The population of a town grows at the rate of $10\%$ per year. Using differential equation, find how long will it take for the population to grow $4$ times.

Answer 1

Since increase in population speeds up with increase in population and let $x$ be the population at any time $t$

$\therefore \frac{dx}{dt}\alpha x$

$\frac{dx}{dt}=rx$

where $r$ is proportionality constant.

Rewriting,

$\frac{dx}{x}=rdt$

Integrating,

$lnx=rt+c$

where $c$ is the a constant

Exponentiating on both sides with $e$

$x=e^{rt+c}=A{e}^{rt}$, where $A=e^c$

Let the initial population $x_0$, then

$x_0=A\dots \left(1\right)$

If the population is increased $4$ times in time $t$, then

$4x_0=A{e}^{rt}\dots \left(2\right)$

From (1), $x=A$

$4x_0=x_0{e}^{rt}$

Taking log on both sides

$\ln 4=rt$

$r=0.1$...(as rate of increase is given as $10\%$)

$t=\frac{\ln 4}{0.1}=\frac{1.38}{0.1}=13.8$ years that is $13$ years and around $10$ months