Answer to Question #282503 in Differential Equations for bilal

Question

The population of a certain state is known to grow at a rate proportional to the number of people presently living in the state. If after 10 years the population has trebled and if after 20 years the population is 150,000, find the number of people initially living in the state

Answer 1

Solution:

Given that the rate of increase of the population of a country is proportional to the number of people living presently in the state.

$$\frac{dP}{dt}=kP$$

where P is the population and k is the proportionally constant.

$$\int \frac{1}{P}dP=\int kdt \implies lnP=kt+c$$

Let P_i be the initial population of the state:

$$t=0 \implies P=P_i$$

$$P=P_ie^{kt}$$

Given at t=10 years population is trebled which gives:

$$3=e^{10k} \implies k=\frac{ln(3)}{10}=0.10986$$

Given at t = 20 years population of the state is 150,000:

$$150.000=P_ie^{20\frac{ln(3)}{10} }$$

$$P_i=\frac{150,000}{9}$$

$$P_i=\frac{5000}{3}=16666.66$$

Answer: the initial population of the state is 16666.66