Which of the following graphs is the solution to the logistic differential equation \( \displaystyle \frac{dy}{dt}=\frac{y}{5}\left(1-\frac{y}{500}\right)\) with the initial condition \(y(0)=100\) ?

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A logistic differential equation has the general form:

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

Since we know the carrying capacity (\(L=500\)) and the \(y\)-intercept (\(y(0)=100\)), we can pick the graph which reflects this.