Question

In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation dP dt = P(a − bP), where a and b are positive constants. Although we will come back to this equation and solve it by an alternative method in Section 3.2, solve the DE this first time using the fact that it is a Bernoulli equation.

Answer 1

Answer:

If  $K$ is a constant of integration, then

$P = {\displaystyle \frac{1}{b/a + Ke^{-at}}}$

Step-by-step explanation:

According to the information of the problem we know that

${\displaystyle \frac{dP}{dt} = P(a-bP) }$

Remember that in general a Bernoulli equation is an equation of the type

$y' + p(x)y = q(x)y^n$

And the idea to solve the equation is to substitute

${ \displaystyle v = y^{1-n}}$

Now for this case

${\displaystyle \frac{dP}{dt} - Pa = -bP^2}$

Then we substitute

$v = P^{1-2} = P^{-1}$

Therefore

$P = v^{-1}$

and if you compute the derivative of that you get that

${\displaystyle \frac{dP}{dt} = -v^{-2} \frac{dv}{dt}}$

Now you substitute that onto the original equation and get

${\displaystyle \frac{dP}{dt} - Pa = -bP^2}$

${\displaystyle -v^{-2} \frac{dv}{dt} - v^{-1} = -bv^{-2}$

If you multiply everything by  $-v^2$  you get that

${\displaystyle \frac{dv}{dt} + v = b }$

That's a linear differential equation and the solution would be

$v = {\displaystyle \frac{b}{a} + Ke^{-at}} = P^{-1}$

Where $K$ is a constant of integration, then

$P = {\displaystyle \frac{1}{b/a + Ke^{-at}}}$