Unit 3 test: Differential Equations

Which differential equation could have the slope field shown?

For the differential equation $\dfrac{dy}{dx} = xy - y^2$, find the slope of the solution curve through the point $(2, -1)$.

Find the particular solution $y = f(x)$ of $\dfrac{dy}{dx} = xy^2$ with $f(0) = 1$.

A cup of tea cools according to $\dfrac{dy}{dt} = -\dfrac{1}{2}(y - 10)$, where $y$ is the temperature in degrees Celsius and $t$ is in hours. If $y(0) = 40$, find $y(t)$.

A drug in the bloodstream is eliminated at a rate proportional to the amount present, with a half-life of 8 hours. A patient starts with 100 mg. How many milligrams remain after 20 hours? Round to the nearest hundredth.

Let $y = f(x)$ solve $\dfrac{dy}{dx} = y - x$ with $f(0) = 2$. Use Euler's method with two steps of equal size to approximate $f(1)$.

Let $y = f(x)$ solve $\dfrac{dy}{dx} = 2x - y$ with $f(1) = 3$. What is the approximation for $f(2)$ found by using Euler's method with two steps of equal size?

Let $y = f(x)$ solve $\dfrac{dy}{dx} = xy + 1$ with $f(0) = 0$. Use Euler's method with step size $0.1$ to approximate $f(0.3)$. Round to three decimal places.

Let $y = f(x)$ solve $\dfrac{dy}{dx} = 3 - y$ with $f(0) = 5$. Euler's method with a positive step size is used to approximate $f(1)$. Which statement is true?

A population satisfies $\dfrac{dP}{dt} = 0.6P - 0.0012P^2$. What is the carrying capacity?

For the population $\dfrac{dP}{dt} = 0.6P - 0.0012P^2$ with $P(0) = 40$, what is the maximum value of $\dfrac{dP}{dt}$?

For the same model, $\dfrac{dP}{dt} = 0.6P - 0.0012P^2$, suppose instead that $P(0) = 700$. Find $\displaystyle\lim_{t \to \infty} P(t)$.

Find the particular solution $P(t)$ of $\dfrac{dP}{dt} = 0.1P\left(1 - \dfrac{P}{400}\right)$ with $P(0) = 50$.

A population $P(t)$ satisfies $\dfrac{dP}{dt} = 0.25P\left(1 - \dfrac{P}{2400}\right)$ with $P(0) = 1800$. Which statement is true for $t \gt 0$?