Answers 1-4 is right but 5 is incorrect. Any help would be great. Because of res

Question

Answers 1-4 is right but 5 is incorrect. Any help would be great.

Because of restoration of an island habitat the maximum population of birds it can support at a time t is given by t/80 1000 E The growth rate of the population p(t) is equal to 1/20 of the difference between the maximum population and the current population. Initially the island has a population of 200 birds. This problem can be written as a differential equation of the form p? + f(t) p = g(t) What is f(t) in this case? f(t) Enter just an expression. What is g(t) in this case? g(t) = Enter just an expression. What is the integrating factor Phi(t) in this case? Phi(t) = Enter just an expression for Phi(t). What did you calculate for the following antiderivative? int g(t) Phi (t) dt = Enter just an expression. What is the solution to the initial value problem? p(t) = Enter just an expression for p(t), do not enter an equation.

Explanation / Answer

From the problem,

p'(t) = [pmax - p(t)]/20 = (1000et/80 )/20- p(t)/20    =>

p'(t) + (1/20) p(t) = 50et/80

Therefore, comparing the above equation to p'(t)+f(t)p(t)= g(t), we have

1. f(t) = 1/20

2. g(t) = 50et/80

3. (t) = ef(t)dt = e1/20dt = et/20

4. g(t)(t)dt = 50et/80et/20dt=50 et/16dt = 50 (16) et/16 = 800 et/16

5. p(t) = g(t)(t)dt/(t) = (800 et/16)/et/20 +C = 800 et/80 +C

When t = 0, we know that p(0) = 200, therefore,

200 = 800 +C     =>   C = -600

Therefore,

p(t) = 800 et/80 - 600

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