Question

Let population of a country be increasing at the rate proportional to its population. If the

population has increased to 25% in 10 years, how long will it take to become double?

Answer 1

Since the population increases at a rate proportional to the population,

$\frac{dP}{dt} =kP,k>0$

Solving the above differential equation,

$\frac{dP}{dt} =kP\\ \frac{dP}{P} =kdt\\ \int \frac{dP}{P} =\int kdt\\ \ln P=kt+C$

Let the initial population be $P_0$. Then at $t=0$,

$\ln P_0=0+C ,C=\ln P_0$

$\ln P=kt+\ln P_0\\ \ln (\frac{P}{P_0})=kt\\ \frac{P}{P_0}=e^{kt}\\ P=P_0e^{kt}$

Also, when $t=10,P=1.25P_0$. So,

$1.25P_0=P_0e^{10k}\\ e^{10k}=1.25$

When the population doubles, $P=2P_0$

$2P_0=P_0e^{kt}\\ 2=e^{kt}\\ t=\frac{\ln 2}{k} \\ t=\frac{\ln 2}{(\ln 1.25)/10} \\ t=31.06$

The population doubles in $31.06 \; years$