Unit 7 test: Differential Equations

A rumor spreads through a school of 500 students. The rate at which the number of students who have heard the rumor, $N$, increases is proportional to the product of the number who have heard it and the number who have not. Which differential equation models $N$, where $k$ is a positive constant?

The temperature $T$ of a cooling object, in degrees Celsius, satisfies $\dfrac{dT}{dt} = -0.15(T - 25)$, with $t$ in minutes. Find $\dfrac{dT}{dt}$ when $T = 105$.

For what value of $k$ is $y = kx^4$ a solution of $x\dfrac{dy}{dx} - 2y = 6x^4$?

Find all values of $r$ for which $y = e^{rx}$ is a solution of $y'' - 5y' + 6y = 0$.

Which function is a solution of $\dfrac{dy}{dx} = y + 1$?

In the slope field for $\dfrac{dy}{dx} = x^2y - 3$, what is the slope of the segment at the point $(2, -1)$?

The slope field below is for which differential equation?

Let $y = f(x)$ be the solution of $\dfrac{dy}{dx} = y^2 - x$ through the point $(1, 2)$. Find the value of $\dfrac{d^2y}{dx^2}$ at that point.

For the differential equation $\dfrac{dy}{dt} = 0.5(8 - y)$, let $y(t)$ be the solution with $y(0) = 2$. What is $\displaystyle\lim_{t \to \infty} y(t)$?

Find the particular solution $y = f(x)$ of $\dfrac{dy}{dx} = 4x^3y$ with $f(0) = 3$.

Find the particular solution $y = f(x)$ of $\dfrac{dy}{dx} = \dfrac{2x}{y}$ with $f(1) = 3$.

Let $y = f(x)$ be the particular solution of $\dfrac{dy}{dx} = y^2$ with $f(0) = \dfrac12$. What is the domain of $f$?

A radioactive substance has a half-life of 8 days. How many milligrams of a 200-milligram sample remain after 20 days? Round to the nearest tenth.

A cake at $180^\circ$F cools in a $72^\circ$F room, and its temperature satisfies $\dfrac{dT}{dt} = -0.04(T - 72)$, where $t$ is in minutes. Find $T(t)$.

A population grows at a rate proportional to its size. It has 500 members at $t = 0$ and 1500 members at $t = 4$ years. How many members will it have at $t = 6$ years? Round to the nearest whole number.