Answer:
[tex]p(t) = Ae^{\frac{kt^2}{2}}[/tex]
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.
[tex]p' = kp[/tex]
where k is a constant.
Solving the given differential equation, we have,
[tex]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}}[/tex]
