Question

The following differential equation is separable as it is of the form dP/dt = g(P)h(t).

Answer 1

Answer:

$P(1-P) = Ae^t$

Step-by-step explanation:

given differential equation is

$\frac{dP}{dt} =P-P^2\\\frac{dP}{dt} = g(P) h(t)\\g(P) = P-P^2\\h(t) = 1$

So this is separable

We separate as

$\frac{dP}{P-P^2} =1dt\\\frac{dP}{P(1-P)} =1dt$

Resolve into partial fractions to solve this

$dP(\frac{1}{P}+ \frac{1}{1-P} ) = 1dt\\ln P(1-P) = t+C\\P(1-P) = Ae^t$

So solution is$P(1-P) = Ae^t$