Thread: separable and exact ODE

I'm unsure of how to set up the differential equation for this problem:

Given that the earth's population on July 1980 was 4,473,000,000 and that on July 1987 it was 5,055,000,000. Assuming that P satisfies model A. When would the earth's population reach 10 billion? Thanks to all in advance for your help!

What is model A?

I'm sorry model A is P' = kP where P' is the first derivative.

Let t represent time measured in years and let July 1980 coincide with . Let represent the population at time t in billions. Using the given model, we have the IVP:

where

Separating the variables in the ODE, we have:



Integrate:







Now, you have two points on the curve, from which you may determine the two constants. Once you have these, then set to answer the question.

So I'm still a little lost. What is the k value? Is it the rate?

I found k = 5.055 - 4.473 = .582
where .582/7 = .0831 so that's the rate of the increase of people per year.

-plug k into the t equation to solve for c1 which gives me .2235

put c1 back into t equation and my answer is not correct. The answer is between mid 2020 and 2035 according to the book. Please show me where i'm going wrong.

We have the two points (0,4.473) and (7,5.055) so this gives us the system:





From the first equation, we find:



Putting this into the second equation, we find:



And so we have:



Now, let and we find:



This is the year 2026.

Cool I was thinking too hard. Thank you again!