First-Order Differential Equations (College Board AP® Calculus BC): Exam Questions

A cylindrical tank contains water and is being used to fill up a garden pond. The tank is standing upright on its circular base. The rate of change of the height $h$ of the water in the tank with respect to time $t$ is modeled by $\frac{dh}{dt}=-\frac{1}{3}\sqrt{h^3}$, where $h$ is measured in feet and $t$ is measured in seconds.

At time $t=0$ seconds, the height of water in the tank is 4 feet. Use separation of variables to find an expression for $h$ in terms of $t$.

Consider the differential equation $\frac{dy}{dx}={\left(y-2\right)}^3\sin \left(\pi x\right)$.

There is a horizontal line with equation $y=c$ that satisfies this differential equation. Find the value of $c$.

Consider the differential equation $\frac{dy}{dx}=\frac{x-6}{y^2}$.

Find the particular solution $y=f\left(x\right)$ to the differential equation with the initial condition $f\left(0\right)=3$.

Consider the differential equation $\frac{dy}{dx}=x^3\left(y+3\right)$.

Describe all the points in the $xy$-plane for which the slopes of tangent lines drawn on a slope field for that differential equation will be negative.

Consider the differential equation $\frac{dy}{dx}=\frac{y^2}{2x-3}$.

Find the particular solution $y=f\left(x\right)$ to the differential equation with the initial condition $f\left(2\right)=2$.

$f$ is a twice-differentiable function for all values of $x$, with $f\left(0\right)=0$. The table above gives values of the derivative of $f$, $f^{\text{'}}$, for selected values of $x$.

Use Euler's method, with two steps of equal size starting at $x=0$, to approximate $f\left(\pi \right)$. Show the computations that lead to your answer.

Consider the logistic differential equation $\frac{dy}{dt}=y\left(4-\frac{y}{3}\right)$ for a population $y$. Let $y=f\left(t\right)$ be the particular solution to the differential equation for a given initial condition.

If $f\left(0\right)=15$, what is the range of $f$ for $t\ge 0$?

If, instead, $f\left(0\right)=2$, for what value of $y$ is the population growing the fastest?

Use separation of variables to find $y=f\left(x\right)$, the particular solution to the differential equation $\frac{\text{d}y}{\text{d}x}=\frac{1}{2}\text{sin}\left(\frac{\pi }{2}x\right)\sqrt{y+7}$ with the initial condition $f\left(1\right)=2$.

Use separation of variables to find $y=A\left(t\right)$, the particular solution to the differential equation $\frac{\text{d}y}{\text{d}t}=\frac{12-y}{3}$ with initial condition $A\left(0\right)=0$.

Consider the differential equation

$\frac{dy}{dx}=\frac{y^2}{x-1}$

On the axes provided, sketch a slope field for the given differential equation at the six points indicated.

At time $t=0$, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is $91 °\text{C}$ at time $t=0$, and the internal temperature of the potato is greater than $27 °\text{C}$ for all times $t>0$. The internal temperature of the potato at time $t$ minutes is modeled by the function $H$ satisfying the differential equation $\frac{dH}{dt}=-\frac{1}{4}(H-27)$, where $H\left(t\right)$ is measured in degrees Celsius and $H\left(0\right)=91$.

Write an equation for the line tangent to the graph of $H$ at $t=0$. Use this equation to approximate the internal temperature of the potato at time $t=3$.

Consider the differential equation $\frac{dy}{dx}=\frac{1}{3}x(y-2{)}^2$.

A slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point $(0,2)$, and sketch the solution curve that passes through the point $(1,0)$.

Find the particular solution $y=f\left(x\right)$ to the differential equation $\frac{\text{d}y}{\text{d}x}=y·\left(x\text{ln}x\right)$ with initial condition $f\left(1\right)=4$.

Consider the differential equation $\frac{dy}{dx}=\frac{8-x^3}{y}$.

Find the particular solution $y=f\left(x\right)$ to the differential equation such that the line $y=-4$ is tangent to the graph of $f$.

Scientists are studying a population of leopards in a nature reserve. The rate of change of the population is proportional to the difference between the current population and the maximum population that the reserve can support. If $P\left(t\right)$ is the population at time $t$ months after the start of the study, then

$\frac{dP}{dt}=\frac{1}{20}\left(90-P\right)$

Use separation of variables to find the particular solution to the differential equation with initial condition $P\left(0\right)=30$. Give your answer as an expression for $P$ in terms of $t$.

Use separation of variables to find $y=f\left(x\right)$, the particular solution to the differential equation $\frac{dy}{dx}=\cos \left(\frac{\pi }{5}x\right)\sqrt{y-2}$ with the initial condition $f\left(5\right)=11$.

Consider the differential equation $\frac{dy}{dx}=x-\frac{1}{2}y+1$.

Find the values of the constants $m$ and $b$, for which $y=mx+b$ is a solution to the differential equation.

Consider the differential equation $\frac{dy}{dx}={\left(y+3\right)}^3\sin \left(\pi x\right)$.

Find the particular solution $y=f\left(x\right)$ to the differential equation with the initial condition $f\left(1\right)=0$

Let $y=f\left(x\right)$ be the particular solution to the differential equation $\frac{dy}{dx}=x{y}^2\ln x$ with initial condition $f\left(1\right)=2$. Use Euler's method, with two equal steps starting at $x=1$, to approximate $f\left(\frac{3}{2}\right)$. Determine whether this approximation is an over- or underestimate, being sure to justify your answer.

A population is modeled by a function $P$ that satisfies the logistic differential equation $\frac{dP}{dt}=4P\left(1-\frac{P}{16}\right)$.

Use separation of variables to find $P\left(t\right)$ if $P\left(0\right)=5$.

Determine the value of the time, $t$, at which the population, $P$, is changing the fastest.

Use separation of variables to find an expression for $M\left(t\right)$, the particular solution to the differential equation $\frac{\text{d}M}{\text{d}t}=\frac{1}{4}(40-M)$ with initial condition $M\left(0\right)=5$.

Use separation of variables to find $y=H\left(t\right)$, the particular solution to the differential equation

$\frac{\text{d}H}{\text{d}t}=\frac{1}{2}(H-1)\text{cos}\frac{t}{2}$

with initial condition $H\left(0\right)=4$.

For $t<10$, an alternate model for the internal temperature of the potato at time $t$ minutes is the function $G$ satisfying the differential equation $\frac{dG}{dt}=-(G-27{)}^{2/3}$, where $G\left(t\right)$ is measured in degrees Celsius and $G\left(0\right)=91$. Find an expression for $G\left(t\right)$. Based on this model, what is the internal temperature of the potato at time $t=3$?