Question

The human population of a small pacific island satisfies the logistic law dx/dt = kx - λx^2, with k=.04, λ=2(10^-7), and time t measured in years.
Find a formula for the population in future years.

Answer 1

I got this far:
please help!

Answer 2

also I have the answer. It should be:

Answer 3

what is initial population?

Answer 4

50,000

Answer 5

ok i think your "B" coefficient is wrong

I think its easier or at least less confusing if you solve the diff equ with the constants then plug in the values later

I get
A = 1/k
B = L/k

Answer 6

\[\int\limits \frac{dx}{x(k-\lambda x)} = \int\limits dt\]
\[\int\limits \frac{A}{x} + \frac{B}{k - \lambda x} = \int\limits dt\]
\[Ak + (B - \lambda A)x = 1\]
\[A = \frac{1}{k}\]
\[B = \lambda A = \frac{\lambda}{k}\]

Answer 7

after integrating you should get
\[\frac{1}{k}[\ln x - \ln (k -\lambda x)] = t + C\]
combine logs and raise everything as power of e
\[\frac{x}{k - \lambda x} = C e^{kt}\]
I would solve for constant of integration here using initial value ,50,000 or 5(10^4)
\[C = \frac{5(10^4)}{4(10^{-2}) - 2(10^{-7})*5(10^4)} = \frac{5 (10^6)}{3}\]
Using algebra and solving for x
\[x = \frac{C k e^{kt}}{1 + C \lambda e^{kt}}\]
plug in values , k, lambda, C ... simplify

Answer 8

thanks so much but can you explain why the book answer has 3 + in the denominator

Answer 9

nevermind I got it THANK YOU