During the period from 1790 to 1920, a country's population P(t) (t in years) grew from 4.1 million to 110.1 million. Throughout this period, P(t) remained close to the solution of the initial value problem:

(dP/dt) = (0.03142) P - (0.0001485) P², P(0) = 3.9

(a) What 1930 population does this logistic equation predict?
(b) What limiting population does it predict?

(a) 128 million

(b) The limiting population is 211.6 million

Step-by-step explanation:

Ordinary  Differential Equation (ODE)

It has been found a country's population P(t) (t in years from 1790) is modeled by the ODE:

With an initial value of P(0)=3.9. Note this value differs from the 4.1 million originally suggested.

This is a variable separable differential equation that can be rewritten as

Integrating

The right-side integral is (for a constant A)

Let's compute the left-hand integral

Factor the denominator

Substitute

Substitute into the integral

Solving

Undo substitution

Solving the ODE

Taking exponentials and using a generic constant K

Solving for P

Use the initial value t=0, P=3.9

We get the value of K

The final solution for P is

(a) Compute P for the year 1930, t=1930-1790=140 years

(b) For larger values of t we evaluate the limit value of P, making

The limiting population is 211.6 million