I have the equation dP/dt = kP(1 - P/A). It is supposed to describe a logistical situatuon involving the carrying capacity of the system.(adsbygoogle = window.adsbygoogle || []).push({});

k is a constant, and A is the carrying capacity of the system. t is time and P is population as a function of time. P(0) = P_{0}. I solved c (the integration constant) to be:

c = -ln|(P_{0})/(A - P_{0})|

I'm trying to solve the equation in terms of t.

In my calculus book, a similar equation is given with an explanation.

dP/dt = 0.1P(1 - P/300)

With an initial condition of P(0) = 50, c is found to be ln(1/5). A = 300 and k = 0.1.

I follow along well up to this point.

After solving for c, the book lists the rearranged equation as:

P(t) = 300/(1 + 5e^{-0.1t})

I don't understand how they went from one equation to the other, the closest I could come with the general equation was:

P(t) = (Ae^{kt}+ Q)/(1 + e^{kt}) where Q = (P_{0})/(A - P_{0})

Which would coincide with an equation of:

(300e^{0.1t}+ .2)/(1 + e^{0.1t})

Which, when graphed, is not equivalent to the equation given by the book.

Can anyone go over how to solve the general equation? I think I'm missing some crucial point.

Thank you!

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# Differential Equations - Logistic Model

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