Can someone help me to complet this , by using the chapter 7.5 on the book :

( integration of rational functions using partial functions)

you can choose the numbers from any source and let me know what it's

Find a situation in your life can be analyzed using our recent methods. Pose a question. Gather the measurements, work through the problem, explain your results. In this collection of sections we have only had one section where I noticed applications. That has been in section 7.5 where we developed the logistic formula for modeling population growth. I anticipate many people modeling different populations, so let me give some clarifying details: You should show me the derivations of your formula. That is where you will be demonstrating the skills we develop in class such as partial fraction decomposition and solving a differential equation. Find some population interesting to you that has at least two data points. Then you will probably need to make up an estimate for a carrying capacity. These models will be more meaningful if done across large populations (cities, regions, the planet) than across households. After you have that part done you will take your scratch work and make a nice write-up. You will probably include diagrams and maybe even a picture. You will have a math-fluent friend look over your work. Then you will bring it with you on test day. EXAMPLE 9 A population grows according to the logistic differential equation y' = 0.0003y(2000 - y). The initial population size is 800. Solve this dif- ferential equation and use the solution to predict the population size at time ( = 2. Section 7.5 Integration of Rational Functions Using Partial Fractions 409 SOLUTION Writing y' as dy/di, we see that the differential equation can be written as dy dt = 0.0003y(2000 - y) dy = 0.0003 dt y(2000 - y) dy y(2000 - y) 0.0003 dt The integral on the left can be evaluated using the method of partial fractions. We write 417 B y(2000 - y) V 2000 - y which leads to 1 = A(2000 - y) + By Substituting y = 0 and y = 2000 yields 1 = 2000 A 1 = 2000 BThus, A = and B = 2000 leading to 2000 2000 y dy = 0.0003t + C 2000(2000 - y) -In y 2000 2000 -In (2000 - y) = 0.0003t + C V In = 0.61 + 2000C 2000 - y y 0.61 +2000C 2000 - y = Ge0.61 2000 - y Here, C1 = e2000. At this point we can use the initial condition y(0) = 800 to determine Cj. 800 = Geo.6-0 2000 - 800 800 G = WIN 1200 Thus : 0.61 WIN 2000 - y y = = (2000 - y )e0.6 4000 y + 3 Ve0.61= - 20.61 3 (4000/3) e 0.61 4000/3 1 + (2/3)e0.6 2/3 + e-0.61