Doomsday Prediction. Doomsday prediction. In 1960, three electrical engineers at the University of Illinois published a paper in Science titled "Doomsday.

Winston Todd 2022-10-23 Answered
Doomsday Prediction
I have a calculus problem I can't seem to figure out. Any help would be appreciated!
Doomsday prediction. In 1960, three electrical engineers at the University of Illinois published a paper in Science titled "Doomsday." Based on world population growth data from 1000 AD to 1960 AD, the engineers found that population growth was faster than proportional to the population size. Using the data, they modeled the growth of the population as
d P / d t = 0.4873 P 2
where P is the population size in billions and t is centuries after 1000 AD.
1) Solve this differential equation.
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Answers (1)

bigfreakystargl
Answered 2022-10-24 Author has 23 answers
Step 1
As you have a "nice form" differential equation here, you can use Separation of Variables and integration to solve. Note that
d P d t = 0.4873 P 2
can be written as
d P P 2 = 0.4873 d t
Now you can integrate both sides
d P P 2 = 0.4873 d t
Thus
1 P + C P = 0.4873 t + C t
Step 2
To finish, note that C P and C t are arbitrary constants due to the indefinite integrals. Thus we can rewrite as
1 P = 0.4873 t + C
which is the same as
P = 1 C 0.4873 t
for some real C. Now as P ( 0 ) = 0.2 , C = 5. So finally we have
P = 1 5 0.4873 t
As the denominator approaches 0, the population approaches . This occurs when 5 0.4873 t = 0. Thus t = 10.2606197.... and as t is the number of centuries after 1000 AD, we can expect an infinite population in 10 years from now.
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