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Calculus

First Order Differential Equations

Concept Quizzes

Differential Equations - Formulate a Statement
Differential Equations - Degree and Order
Differential Equations - Euler's Method - Step size of 1
Differential Equations - dy/dx = f(x)
Differential Equations - dy/dx = g(y)
Differential Equations - Variable Separable
Logistic Differential Equations
Differential Equations - Modeling

Challenge Quizzes

First Order Differential Equations: Level 2 Challenges

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 $P(t)=100?$

\displaystyle{t = \frac{1}{3} \ln \frac{583}{87}}

\displaystyle{t = \frac{1}{3} \ln \frac{583}{85}}

\displaystyle{t = \frac{1}{3} \ln \frac{583}{95}}

\displaystyle{t = \frac{1}{3} \ln \frac{583}{100}}

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by Brilliant Staff

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

\displaystyle{P(t) = \frac{8 P_{0}}{P_{0} + (5+P_{0})e^{-5 t}}}

\displaystyle{P(t) = \frac{8 P_{0}}{P_{0} - (5-P_{0})e^{-8 t}}}

\displaystyle{P(t) = \frac{8 P_{0}}{P_{0} + (8-P_{0})e^{-5 t}}}

\displaystyle{P(t) = \frac{5 P_{0}}{P_{0} - (8+P_{0})e^{-8 t}}}

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by Brilliant Staff

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 $P(0)$ ?

\displaystyle{\frac{448}{85}}

\displaystyle{\frac{448}{87}}

\displaystyle{\frac{2240}{437}}

\displaystyle{\frac{2240}{433}}

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by Brilliant Staff

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 $P(4)?$

\frac{63}{22}

\frac{42}{13}

\frac{14}{5}

\frac{63}{23}

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by Brilliant Staff

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 $P(\ln3)?$

\displaystyle{\frac{378}{53}}

\displaystyle{\frac{42}{5}}

\displaystyle{\frac{378}{55}}

\displaystyle{\frac{126}{19}}

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by Brilliant Staff