Question

Consider the growth of a population p(t). It starts out with p(0)=A. Suppose the growth is unchecked, and hence p′=kp for some constant k. Then p(t)=

Answer 1

`p(t) = Ae^{(kt^2)/(2)}`

Explanation:

We are given the following information in the question:

The growth of population is given by the function p(t).

p(0)=A where A is a constant.

`p' = kp`

where k is a constant.

Solving the given differential equation, we have,

`p' = kp\\\\\displaystyle(dp)/(dt) = kp\\\\(dp)/(p) = kt~ dt\\\\\text{Integrating both sides}\\\\\int (dp)/(p) = \int kt~ dt\\\\\log p = (kt^2)/(2) + C\\\\\text{where C is the integration constant}\\\\\text{Putting t = 0}\\\\\log p_0 = C\\\\\log p = (kt^2)/(2) + \log p_0\\\\\log p - \log A = (kt^2)/(2)\\\\\log (p)/(A) = (kt^2)/(2)\\\\p(t) = Ae^{(kt^2)/(2)}`