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

Her `dx/dy prop x` [Since the increase in population speeds up with the increase in population] and let x be the population at any time t.

`:. dx/dy = rx`   (where r is proportionality constant)

`:. dx/ x = r.dt`

integrating both sides

In x = rt + c,         (where c is the integration constant)

`:. x = e^(rt + c)`

`x = Ke^(rt)` where `K = e^c`

Here r is the rate of increase and K is the initial population let x₀ then t = 0

`x_0 = ke^o => k = x_0`

Given to find the time t taken to attain 4 times population, so `x = 4x_0`

So, `x = Ke^(rt)`

`=> 4x_0 = x_0e^0.10t`

`2 = e^0.05t`

Taking log on both sides

In 2 = In `e^(0.1t)`

0.69314 = 0.1t        t = 6.9314