Question

During the period from 1790 to 1930, the US population P(t) (t in years) grew from 3.9 million to 123.2 million. Throughout this period, P(t) remained close to the solution of the initial value problem.

(dP/dt)= 0.03135P – 0.0001489P^2 , P(0) = 3.9

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

Ps: I have answered this question and got a result, hence, I'm just checking my work, so please provide a detailed answer. THANKS!!!!

Answer 1

Explanation:

Given that during the period from 1790 to 1930, the US population P(t) (t in years) grew from 3.9 million to 123.2 million. Throughout this period, P(t) remained close to the solution of the initial value problem.

`(dP)/(dt) =0.03135P =0.0001489P^2, P(0) = 3.9`

a) 1930 population is the population at time t = 40 years taking base year as 40

We can solve the differential equation using separation of variables

`(dP)/(0.03135P – 0.0001489P^2 ) =dt\\(dP)/(-P(0.03135 – 0.0001489P ) =dt`

Resolve into partial fractions

`(31.8979)/(P) -(0.00474)/(0.0001489P-0.03135)`

Integrate to get

ln P -0.00474/0.0001489 (ln (0.0001489P-0.03135) = t+C

ln P -31.833 (ln (0.0001489P-0.03135) =t+C

`(P)/(( ( (0.0001489P-0.03135)^(31.833) ) =Ae^t`

Limiting population would be infinity.