How Can You Solve the US Population Model Using Differential Equations?

In summary, the conversation discusses solving a differential equation in terms of the constants a and b. The equation is (1/P)(dP/dt) = ax + b, where a = -0.0001 and b = 0.0338. The goal is to find the function P(t) using separation of variables and an initial condition of P0 = 3.9. The final solution is P = 3.9 e^(0.0338t - 0.0001t^2/2), which works well for the first 100 years but may not be accurate for longer periods of time.
  • #1
pjallen58
12
0
Use P0 = 3.9 (1790 population) as your initial condition to find the particular solution for this differential equation. Note: You may find it easier to solve in terms of the constants a and b. Show all the steps in your solution.

This is the last step to a multi-part problem. I basically did a scatter plot of the relative growth rate by dividing an approximate growth rate by the US population between 1790 and 2000, did a linear regression and obtained the equation below.


y = -.0001x + .0338 and (1/P)(dP/dt) = ax + b

I do not exactly know what they are asking to be done. Do they want me to solve for P'(t)? If so, how does the 1/P on the left side of the equation effect the right side? Any help would be appreciated. Thanks.
 
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  • #2
P'? They want you to solve for P, it's a differential equation. Remember that the x-axis is time, and then it's separation of variables.
 
  • #3
Thanks for the reply. I already have all the P's. If you could give a little more information to clear things up it would be appreciated. Thanks.
 
  • #4
You typed (1/P)(dP/dt) = ax + b


Where a is -0.0001 and b is .0338 and x should be time

So you haven't found the function P(t) yet, but there is the equation. After you separate variables you'd have

dP/P=(a*t+b)*dt, then you need to integrate and use your initial condition
 
  • #5
Thanks. After I sent the last reply it clicked that I understood what you ment by solve for P as the variable. Thanks again.
 
  • #6
Here is what I calculated:

Use P0 = 3.9 as your initial condition to find the particular solution for this differential equation.

(1/P)(dP/dt) = b + at

y = -.0001t + .0338

dP/P = (.0338 - .0001t)dt

ln P = .0338t - (.0001t^2/2) + C

At t = 0, P0 = 3.9 so then C = ln 3.9

ln P = .0338t - (.0001t^2/2) + ln 3.9

Take the exponential of both sides,

P = 3.9 e^.0338t-(.0001t^2/2)

This formula seems to work well for the first 100 years but gets out of control after that so not sure if I have everything right. Let me know. Thanks.
 

Related to How Can You Solve the US Population Model Using Differential Equations?

1. How is the US population currently growing or declining?

The US population is currently growing, but at a slower rate than in previous years. According to the US Census Bureau, the estimated population growth rate for 2021 is 0.58%, which is the lowest it has been since 1900.

2. What factors contribute to changes in the US population?

There are several factors that contribute to changes in the US population, including birth rates, death rates, immigration, and emigration. These factors can vary from year to year and can also be influenced by social, economic, and political factors.

3. How is the US population expected to change in the future?

According to the US Census Bureau, the US population is projected to continue growing, reaching 390 million by 2050. However, the rate of growth is expected to slow down as the population ages and birth rates decline.

4. How does the US population compare to other countries?

The US has the third largest population in the world, behind China and India. However, the US has a much lower population density compared to these countries, with a larger land area and a smaller population.

5. How accurate are population models for predicting future population changes?

Population models are based on various assumptions and data, so they can provide a general idea of future population changes. However, they are not always completely accurate as they cannot account for unexpected events or changes in social or economic trends. Therefore, population models should be used as a guide rather than a definitive prediction.

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