# 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. - Mathematics

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.

#### Solution

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

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