Answer to Question #328766 in Differential Equations for lota

Question

The population of a city increases at a rate proportional to the present number. It has an initial population of 50000 that increases by 15% in 10 years. What will be the population in 30

years?

Answer 1

Define $x(t)$ - population of the city at the moment $t$ (in years).

$\frac{dx}{dt}=mx$ - reflect the fact that population of the city increases at a rate proportional to the present number. $m$ is proportional coefficient. Let’s solve this equation:

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

$\ln |x|=mt+\ln C$

$x=Ce^{mt}$ , where $C=const$ .

$x(0)=C=50000$ ;

$x(10)=50000+0.15\cdot50000=57500$

$x(10)=50000e^{10m}=57500$

$e^{10m}=\frac{57500}{50000}=1.15$

$10m=\ln{1.15}$

$m=\frac{\ln{1.15}}{10}\approx0.014$

$x(t)=50000e^{\frac{\ln{1.15}}{10}t}$

Population in 30 years:

$x(30)=50000e^{\frac{\ln{1.15}}{10}\cdot30}=50000\cdot(1.15)^3=$ $76043.75\approx76044$