Fllowing 1st order ODE describing the growth and decay of a population. a) Deter
ID: 1818704 • Letter: F
Question
Fllowing 1st order ODE describing the growth and decay of a population.
a) Determine if the equation is linear or non-linear and autonomous or non-autonomous. If linear, determine if the differential equation is homogeneous or non-homogeneous.
c) At what time will the population reach its maximum? Assuming the initial population is 5 million, what will the maximum population be?
d) According to this model, will the population every reach zero? Explain. Determine how long will it take for the population to reduce to 1 organism?
Explanation / Answer
dP/dt >0 for t<2
<0 for t>0
=> It is non autonomous
Variable separation and integrating
ln p = 2t - t2/2 + ln c
P = C e2t - t^2/2
c= Po
P(t) = Poe(2t - t^2/2)
Max population
dp/dt = 0
2 = t
Max P = 5 e2 million = 36.95 m
For P->0
-(2t - t^2/2) ->
t2/2 - 2t ->
t2/2 - 2t is an increasing function for t>2 => It can reach 0 at an infinitely high value of t
P= 1
=>e(2t - t^2/2) =1/5
t2/2 - 2t =1.61
solve for t
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