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Rudiy27
2 years ago
10

The differential equation in Example 3 of Section 2.1 is a well-known population model. Suppose the DE is changed to dP dt = P(a

P − b), where a and b are positive constants. Discuss what happens to the population P as time t increases.
Mathematics
1 answer:
LuckyWell [14K]2 years ago
6 0

Answer:

Decreases

Step-by-step explanation:

We need to determine the integral of the DE;

dP/dt=P(aP-b)

dP=P(aP-b)dt

1/(dP^2-bP)dP=dt

We can solve this by integration by parts on the left side. We expand the fraction 1/P²:

1/(d-b/P)\cdot{P^2} dP

let

u=d-b/P

du/dP=b/P^2

dP=\int\limits {P^2/b} \, du

P=lnu/b

Substitute u in:

P=ln(d-b/P)/b

Therefore the equation is:

ln(d-b/P)/b=t

We simplify:

d-b/P=e^b^t

P=b/(d-e^b^t)

As t increases to infinity P will decrease

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