The Malthus Equation

Let us go back to the population model that we developed in the section on the exponential function.

$${\frac{dP}{dt}}$$ = kP,        P(0) = P₀

where k is a constant.

This is a separable equation and we can rearrange it to get

$$\int$$$${\frac{dP}{P}}$$ = $$\int$$k dt + C

or

ln|P|= kt + C

Exponentiate both sides and get

P(t) = ±e^Ce^kt

now impose the initial condition P(0) = P₀ and get the result

P(t) = P₀e^kt