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# In this exercise, the logistic equation models the growth of a population. Use the equation to write a logistic differential equation that has the solution $P(t)$.$P(t)=\dfrac{5000}{1+39e^{-0.2t}}$

Solution

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We will start this exercise by comparing the logistic equation to the standard shape of the logistic equation:

$P(t)=\frac{L}{1+be^{-kt}}$

and taking out the constants $k$, $L$ and $b$.

After that, we will write the differential logistic equation by plugging these values into the standard shape of the logistic differential equation:

$\frac{dP}{dt}=kP\left(1-\frac{P}{L}\right)$

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