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# First Order Differential Equations

These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

# Logistic Differential Equations

Given the logistic differential equation $\frac{dP}{dt} = 3 P \left( 1 - \frac{P}{600}\right),$ and $$P(0)=P_{0} = 17,$$ what time $$t$$ satisfies $$P(t)=100?$$

Let $$P_0$$ denote the initial value $$P(0)$$ of the function $$P(t).$$ Which of the following functions satisfy $\frac{dP}{dt} = 5 P \left( 1 - \frac{P}{8}\right)?$

Consider the logistic differential equation $\frac{dP}{dt} = -\ln 2 \times P \left( 1 - \frac{P}{5}\right).$ If $$P(4) = 28,$$ what is the value of $$P(0)$$?

Given the logistic differential equation $\frac{dP}{dt} = -\ln 2 \times P \left( 1 - \frac{P}{18}\right),$ if $$P(0)=P_{0} = 14,$$ what is the value of $$P(4)?$$

Given the logistic differential equation $$\displaystyle{ \frac{dP}{dt} = -2 P \left( 1 - \frac{P}{21}\right)},$$ if $$P(0)=P_{0} = 18,$$ what is the value of $$P(\ln3)?$$

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