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 $y(0)=100$ ?

A logistic differential equation has the general form:

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