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

Answer:

$p(t) = Ae^{\frac{kt^2}{2}}$

Step-by-step 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,

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