AP Calculus AB Unit 7: Differential Equations

Writing differential equations from a description, checking a proposed solution, sketching and reasoning with slope fields, separation of variables for general and particular solutions, and exponential growth and decay models $\frac{dy}{dt}=ky$.

CED topics in this unit: 7.1, 7.2, 7.3, 7.4, 7.6, 7.7, 7.8.

Key formulas and facts

• Separable: $\frac{dy}{dx}=g(x)h(y)$ gives $\int\frac{dy}{h(y)}=\int g(x)\,dx$
• $\frac{dy}{dt}=ky$ gives $y=y_0e^{kt}$
• 'The rate of change of $y$ is proportional to $y$' means $\frac{dy}{dt}=ky$
• Slope field: at each point $(x,y)$ draw a short segment with slope $\frac{dy}{dx}$ evaluated there
• Second derivative from a differential equation: differentiate $\frac{dy}{dx}$ implicitly, then substitute $\frac{dy}{dx}$

Worked examples

• Check a solution: $y=3e^{2x}-1$ gives $y'=6e^{2x}=2(y+1)$, so it solves $\frac{dy}{dx}=2y+2$
• Separate: $\frac{dy}{dx}=\frac xy$ with $y(0)=3$: $\frac{y^2}{2}=\frac{x^2}{2}+C$, $C=\frac92$, so $y=\sqrt{x^2+9}$ (the positive root, because $y(0)>0$)
• Growth: a culture doubles every $5$ hours, so $y=y_0e^{kt}$ with $k=\frac{\ln2}{5}$
• Slope field: for $\frac{dy}{dx}=x-y$, the slopes are $0$ along the line $y=x$
• Concavity from the equation: $\frac{dy}{dx}=xy$ gives $\frac{d^2y}{dx^2}=y+x\frac{dy}{dx}=y+x^2y$

Common traps

• Adding the constant only after exponentiating: $e^{x+C}=Ae^x$, not $e^x+C$.
• Not choosing the sign of a square root, or ignoring the domain of the particular solution.
• Integrating $\frac{dy}{dx}=xy$ as $y=\frac{x^2}{2}y$: separate the variables first.
• Swapping $x$ and $y$ when evaluating slopes for a slope field.
• Losing the constant of integration: without it the initial condition cannot be used.

Practice questions

43 multiple-choice and 4 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

The temperature $T$, in degrees Celsius, of a cup of tea changes at a rate proportional to the difference between $T$ and the room temperature of $21^\circ$C. Which differential equation models this, where $k$ is a constant?

1. $\dfrac{dT}{dt}=kT-21$
2. $\dfrac{dT}{dt}=k(t-21)$
3. $\dfrac{dT}{dt}=k(T-21)$
4. $\dfrac{dT}{dt}=21kT$

Question 2 · Multiple choice

Which function is a solution of the differential equation $\dfrac{dy}{dx}=3y-6$?

1. $y=5e^{3x}-2$
2. $y=2+5e^{3x}$
3. $y=2+e^{x/3}$
4. $y=3x^2-6x$

Question 3 · Multiple choice

Consider the slope field for $\dfrac{dy}{dx}=x(y-1)$. Which statement is true?

1. The slopes are $0$ only along the line $y=1$
2. Every slope in the first quadrant is positive
3. The slopes are $0$ at every point on the $y$-axis and on the line $y=1$
4. The slopes depend on $x$ only

Question 4 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=y^2-4x$ through the point $(1,3)$. What is $\dfrac{d^2y}{dx^2}$ at $(1,3)$?

1. $5$
2. $26$
3. $30$
4. $-4$

Question 5 · Multiple choice

Which is the general solution of $\dfrac{dy}{dx}=2xy^2$ (for $y\ne0$)?

1. $y=\dfrac{1}{x^2+C}$
2. $y=-\dfrac{1}{x^2+C}$
3. $y=Ce^{x^2}$
4. $y=-\dfrac{1}{x^2}+C$

Question 6 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=(y+1)\cos x$ with $f(0)=2$. What is $f\!\left(\dfrac{\pi}{2}\right)$?

1. $2 e$
2. $3 e$
3. $-1 + 3 e$
4. $1 + e$

Question 7 · Multiple choice

A radioactive substance decays at a rate proportional to the amount present. A sample of $80$ grams decays to $50$ grams in $6$ years. To the nearest thousandth of a year, how long after the start will $20$ grams remain?

1. $12.000$
2. $17.697$
3. $8.849$
4. $3.673$

Question 8 · Multiple choice

A population satisfies $\dfrac{dP}{dt}=0.04P$ with $P(0)=1500$, where $t$ is in years. At what time is the population increasing at $120$ people per year?

1. $25.000$
2. $34.657$
3. $17.329$
4. $3000.000$

Question 9 · Free response (9 points)

Consider the differential equation $\dfrac{dy}{dx}=x(y-3)$. Let $y=f(x)$ be the particular solution with $f(1)=4$.

1. (a) Find the slope given by the differential equation at each of the points $(-1,2)$, $(0,2)$, $(1,2)$, $(-1,4)$, $(0,4)$ and $(1,4)$, as you would to sketch a slope field. [1 point]
2. (b) Write an equation of the line tangent to the graph of $f$ at $x=1$, and use it to approximate $f(1.2)$. [2 points]
3. (c) Find $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$. Is the approximation in part (b) an overestimate or an underestimate? Give a reason. [2 points]
4. (d) Use separation of variables to find $y=f(x)$. [4 points]

Question 10 · Free response (9 points)

The rate at which the mass $M$ of a medicine in a patient's bloodstream decreases is proportional to $M$. At time $t=0$ hours, $M=200$ milligrams, and at $t=3$ hours, $M=140$ milligrams.

1. (a) Write a differential equation for $M$ in terms of $t$ and a constant $k$. [1 point]
2. (b) Find $M$ as a function of $t$, and find the value of $k$. [4 points]
3. (c) At what time will $50$ milligrams remain? Give your answer to the nearest thousandth of an hour. [2 points]
4. (d) Find $\dfrac{dM}{dt}$ at $t=3$, and explain its meaning in context. [2 points]

Question 11 · Multiple choice

The rate of change of a quantity $y$ with respect to time $t$ is proportional to the square root of $y$. When $y=9$, $\dfrac{dy}{dt}=6$. Which differential equation models $y$?

1. $\dfrac{dy}{dt}=2\sqrt{y}$
2. $\dfrac{dy}{dt}=6\sqrt{y}$
3. $\dfrac{dy}{dt}=\dfrac23y$
4. $\dfrac{dy}{dt}=\dfrac{y^2}{9}$

Question 12 · Multiple choice

A pie cools in a room kept at $70^\circ$F. The rate of change of its temperature $T$ is proportional to the difference between $T$ and the room temperature. When $T=150$, the temperature is falling at $8$ degrees per minute. Which differential equation models $T$?

1. $\dfrac{dT}{dt}=-8(T-70)$
2. $\dfrac{dT}{dt}=0.1(T-70)$
3. $\dfrac{dT}{dt}=-0.1(T-70)$
4. $\dfrac{dT}{dt}=-0.1T$

Question 13 · Multiple choice

The slope of a curve at each point $(x,y)$ is proportional to the product of $x$ and $y$, and the slope at $(1,4)$ is $2$. Which differential equation describes the curve?

1. $\dfrac{dy}{dx}=2xy$
2. $\dfrac{dy}{dx}=\dfrac{x+y}{2}$
3. $\dfrac{dy}{dx}=\dfrac{xy}{2}$
4. $\dfrac{dy}{dx}=\dfrac{x}{2y}$

Question 14 · Multiple choice

Water drains from a tank so that the depth $h$ decreases at a rate proportional to $\sqrt{h}$. When $h=16$ feet, the depth decreases at $2$ feet per hour. Which differential equation models $h$?

1. $\dfrac{dh}{dt}=-2\sqrt h$
2. $\dfrac{dh}{dt}=-\dfrac{\sqrt h}{2}$
3. $\dfrac{dh}{dt}=\dfrac{\sqrt h}{2}$
4. $\dfrac{dh}{dt}=-\dfrac{h}{8}$

Question 15 · Multiple choice

The rate of change of a quantity $A$ is inversely proportional to $A$. When $A=5$, $\dfrac{dA}{dt}=2$. Which differential equation models $A$?

1. $\dfrac{dA}{dt}=\dfrac{2}{5A}$
2. $\dfrac{dA}{dt}=\dfrac{10}{A}$
3. $\dfrac{dA}{dt}=\dfrac{2A}{5}$
4. $\dfrac{dA}{dt}=\dfrac{5}{2A}$

Question 16 · Multiple choice

Which function is a solution of the differential equation $\dfrac{dy}{dx}=2y$?

1. $y=2e^{x}$
2. $y=e^{2x}$
3. $y=e^{x}+2$
4. $y=x^2$

Question 17 · Multiple choice

Which function is a solution of the differential equation $y''+y=0$?

1. $y=e^{x}$
2. $y=x^2$
3. $y=e^{-x}$
4. $y=3\sin x$

Question 18 · Multiple choice

For every constant $C$, the function $y=Ce^{-3x}+2$ is a solution of which differential equation?

1. $\dfrac{dy}{dx}=-3y+2$
2. $\dfrac{dy}{dx}=3y-6$
3. $\dfrac{dy}{dx}=-3y$
4. $\dfrac{dy}{dx}=-3y+6$

Question 19 · Multiple choice

For which values of $k$ does $y=e^{kx}$ satisfy $y''-5y'+6y=0$?

1. $k=-2$ and $k=-3$
2. $k=2$ and $k=3$
3. $k=1$ and $k=6$
4. $k=5$ only

Question 20 · Multiple choice

For every constant $C$, the function $y=x^2+\dfrac{C}{x}$ is a solution of which differential equation for $x>0$?

1. $xy'-y=x^2$
2. $y'=2x$
3. $xy'+y=3x^2$
4. $xy'+y=x^2$

Question 21 · Multiple choice

Which function satisfies $\dfrac{dy}{dx}=2xy$ and $y(0)=3$?

1. $y=3e^{x^2}$
2. $y=e^{x^2}+2$
3. $y=3e^{2x}$
4. $y=e^{x^2}$

Question 22 · Multiple choice

The function $y=Ae^{2x}-1$ is a solution of $\dfrac{dy}{dx}=2y+2$ with $y(0)=4$. What is $A$?

1. $4$
2. $3$
3. $5$
4. $2$

Question 23 · Multiple choice

Which differential equation produces the slope field in the figure?

1. $\dfrac{dy}{dx}=y-x$
2. $\dfrac{dy}{dx}=x-y$
3. $\dfrac{dy}{dx}=x+y$
4. $\dfrac{dy}{dx}=xy$

Question 24 · Multiple choice

The figure shows a slope field. Which differential equation does it represent?

1. $\dfrac{dy}{dx}=x+y$
2. $\dfrac{dy}{dx}=-\dfrac{xy}{2}$
3. $\dfrac{dy}{dx}=\dfrac{y}{2}$
4. $\dfrac{dy}{dx}=\dfrac{xy}{2}$

Question 25 · Multiple choice

In the slope field for $\dfrac{dy}{dx}=(x+1)(y-2)$, along which lines are all the segments horizontal?

1. $x=1$ and $y=-2$
2. $y=-1$ and $x=2$
3. $y=x+3$
4. $x=-1$ and $y=2$

Question 26 · Multiple choice

When sketching a slope field for $\dfrac{dy}{dx}=x^2-2y$, what slope should the short segment at $(3,2)$ have?

1. $5$
2. $7$
3. $1$
4. $-5$

Question 27 · Multiple choice

Which differential equation could have generated the slope field shown?

1. $\dfrac{dy}{dx}=y-1$
2. $\dfrac{dy}{dx}=1-x$
3. $\dfrac{dy}{dx}=1-y$
4. $\dfrac{dy}{dx}=x-1$

Question 28 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=y(3-y)$ that passes through $(0,4)$. Which describes the graph of $f$ at $x=0$?

1. decreasing and concave up
2. decreasing and concave down
3. increasing and concave up
4. increasing and concave down

Question 29 · Multiple choice

The slope field shown is for $\dfrac{dy}{dx}=x-y$. A solution curve passes through one of the following points with a horizontal tangent there, and has a relative minimum at that point. Which point?

1. $(1,0)$
2. $(0,1)$
3. $(1,1)$
4. $(2,1)$

Question 30 · Multiple choice

For $\dfrac{dy}{dx}=x+y$, which line is itself a solution curve?

1. $y=-x-1$
2. $y=-x$
3. $y=x+1$
4. $y=-x+1$

Question 31 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=y-2$ with $f(0)=3$. Which describes $f$ for $x>0$?

1. decreasing and concave up
2. increasing and concave up
3. increasing and concave down
4. constant

Question 32 · Multiple choice

What is the general solution of $\dfrac{dy}{dx}=xy$? ($C$ is a constant.)

1. $y=e^{x^2/2}+C$
2. $y=Ce^{x^2}$
3. $y=\frac{x^2}{2}y+C$
4. $y=Ce^{x^2/2}$

Question 33 · Multiple choice

Which relation gives the general solution of $\dfrac{dy}{dx}=\dfrac{x}{y}$? ($C$ is a constant.)

1. $y^2-x^2=C$
2. $y^2+x^2=C$
3. $y-x=C$
4. $y^2-2x^2=C$

Question 34 · Multiple choice

Which relation gives the general solution of $\dfrac{dy}{dx}=e^{x-y}$? ($C$ is a constant.)

1. $e^{y}-e^{x}=C$
2. $e^{y}+e^{x}=C$
3. $y-x=C$
4. $e^{-y}-e^{x}=C$

Question 35 · Multiple choice

Separating variables in $\dfrac{dy}{dx}=\cos x\,(1+y^2)$ leads to which general solution? ($C$ is a constant.)

1. $\arctan y-\sin x=C$
2. $\arctan y+\sin x=C$
3. $\ln(1+y^2)-\sin x=C$
4. $y+\frac{y^3}{3}-\sin x=C$

Question 36 · Multiple choice

Which relation gives the general solution of $\dfrac{dy}{dx}=-2xy^2$? ($C$ is a constant.)

1. $\dfrac1y+x^2=C$
2. $\ln|y|+x^2=C$
3. $\dfrac1y-x^2=C$
4. $y^3+3x^2=C$

Question 37 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=2xy$ with $f(0)=3$. What is $f(1)$?

1. $3 e$
2. $3$
3. $e$
4. $3 e^{2}$

Question 38 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=\dfrac{3y}{x}$ for $x>0$ with $f(1)=2$. What is $f(2)$?

1. $12$
2. $8$
3. $16$
4. $6$

Question 39 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=\dfrac{x^2}{y}$ with $f(0)=2$. What is $f(3)$?

1. $\sqrt{13}$
2. $22$
3. $\sqrt{22}$
4. $3 \sqrt{2}$

Question 40 · Multiple choice

Let $y=f(x)$ be the solution of $\dfrac{dy}{dx}=(y-1)\cos x$ with $f(0)=3$. What is $f\!\left(\frac{\pi}{2}\right)$?

1. $1 + 2 e$
2. $2 e$
3. $3 e$
4. $3$

Question 41 · Multiple choice

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

1. $\ln{\left(4 \right)}$
2. $4$
3. $\ln{\left(5 \right)}$
4. $e^{4}$

Question 42 · Multiple choice

A population satisfies $\dfrac{dP}{dt}=0.05P$, with $P(0)=200$. What is $P(10)$?

1. $329.744$
2. $300.000$
3. $210.254$
4. $543.656$

Question 43 · Multiple choice

A radioactive substance decays according to $\dfrac{dy}{dt}=ky$ and has a half-life of $5$ years. What is $k$?

1. $\frac{\ln{\left(2 \right)}}{5}$
2. $- \frac{\ln{\left(2 \right)}}{5}$
3. $- 5 \ln{\left(2 \right)}$
4. $- \frac{1}{10}$

Question 44 · Multiple choice

A bacteria population grows at a rate proportional to its size and doubles every $3$ hours. If it starts at $100$, what is it after $12$ hours?

1. $800$
2. $400$
3. $1600$
4. $3200$

Question 45 · Multiple choice

The temperature of an object satisfies $\dfrac{dT}{dt}=-0.1(T-70)$, with $T(0)=200$ degrees Fahrenheit and $t$ in minutes. What is $T(10)$?

1. $70.000$
2. $117.824$
3. $130.000$
4. $47.824$

Question 46 · Free response (9 points)

Consider the differential equation $\dfrac{dy}{dx}=x^2(y-2)$. Let $y=f(x)$ be the particular solution with $f(1)=3$.

1. (a) Find the slopes that a slope field for this equation would show at the points $(-1,1)$, $(0,1)$, $(1,1)$, $(-1,3)$, $(0,3)$ and $(1,3)$. [1 point]
2. (b) Write an equation for the line tangent to the graph of $f$ at $x=1$, and use it to approximate $f(1.1)$. [2 points]
3. (c) Find $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$. Is the approximation in part (b) an overestimate or an underestimate? Give a reason. [2 points]
4. (d) Use separation of variables to find $y=f(x)$. [4 points]

Question 47 · Free response (9 points)

The number of bacteria $B(t)$ in a culture, $t$ hours after it is started, grows at a rate proportional to $B$. At $t=0$ there are $500$ bacteria, and at $t=4$ there are $2000$.

1. (a) Write a differential equation for $B$, and find the constant of proportionality $k$. [2 points]
2. (b) Find $B(6)$. [2 points]
3. (c) At what time will there be $10000$ bacteria? Give your answer to the nearest thousandth of an hour. [2 points]
4. (d) Find the rate at which the number of bacteria is increasing at $t=6$. Indicate units of measure. [3 points]