(Population Growth) Over a short interval of time, population grows at a rate di

Question

(Population Growth) Over a short interval of time, population grows at a rate directly proportional to the Size of the population at that time, if P(t) represents the population at time t, then this principle states that: dp/dt kP(t), where k is called the population growth constant. Solve this differential equation to obtain a family of explicit solutions for P(t). (You do not need to know the value of k to do this.) If the initial population is 2,000,000 (i.e. P(0) = 2,000,000), find a particular solution for P(t). (Your answer will still be in terms of the unknown k)

Explanation / Answer

dP/dt = kP(t)

i.

dP(t)/dt = kP(t)
(1/P(t)) dP(t) = k dt
(1/P(t)) dP(t) = k dt
ln (P(t)) = kt + C
P(t) = exp(kt + C)
P(t) = C*exp(kt) = Cekt

ii.

P(0) = 2000000
P(0) = Cek*0
P(0) = C = 2000000
P(t) = 2000000*ekt

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