AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/7/7-1)
Unit 7 · Topic 7.1
7.1 Modeling Situations with Differential Equations
A differential equation is an equation that involves a derivative, like dy/dt = 0.3y. This topic is about turning a sentence that describes how something changes into a differential equation, and reading what an equation tells you about the rate of change.
Key terms
- differential equation
- rate of change
- proportional
- constant of proportionality
- model
What a differential equation is
Most equations you've solved have a number as the answer. A differential equation (often shortened to DE) relates an unknown function to one or more of its derivatives, so its answer is a function. For example, dy/dx = 2x says “the slope of y is always twice x.” The function y = x² fits, and so does y = x² + 5.
In this course, nearly all differential equations are first-order: they involve only the first derivative, dy/dx or dy/dt, and give it as an expression in x and/or y. The independent variable is often t (time), but problems also use other letters, like P for population or V for volume. Read the names carefully.
Translating words into equations
The key phrase is “proportional to.” If A is proportional to B, then A = kB for some constant k, called the constant of proportionality. “The rate of change of y” means dy/dt (or dy/dx).
| Words | Differential equation |
|---|---|
| The rate of change of y is proportional to y | dy/dt = ky |
| The rate of change of P is proportional to the square root of P | dP/dt = k√P |
| The temperature T changes at a rate proportional to the difference between T and 70 | dT/dt = k(T − 70) |
| The rate of change of y is jointly proportional to y and to (100 − y) | dy/dt = ky(100 − y) |
| The volume V decreases at a rate proportional to its surface area S | dV/dt = −kS (with k > 0) |
Signs and constants
If the quantity is growing, the derivative is positive. If it's shrinking, the derivative is negative. You can show this either by stating k > 0 or k < 0, or by writing a minus sign and keeping k positive. Either is fine if you're clear.
You can find k when you're told the rate at one moment. If dP/dt = k√P and the population grows by 30 per year when P = 100, then 30 = k√100 = 10k, so k = 3.
A differential equation tells you slopes right away
You don't need to solve a differential equation to use it. Plug in a point and you get the rate of change there. If dy/dx = x − 2y and the solution passes through (1, 0), the slope there is 1 − 0 = 1, so y is increasing at that point. You can also find a tangent line, or differentiate the equation again to find concavity (7.4). Exam questions use this a lot.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Writing the equation and finding k
A population P grows at a rate proportional to the square root of P. When P = 100, it is growing at 30 animals per year. Write a differential equation for P, find k, and find the growth rate when P = 400.
Show the solutionHide the solution
- Step 1: “Rate of change of P” is dP/dt. “Proportional to the square root of P” means k√P. So dP/dt = k√P.
- Step 2: Use the given rate: 30 = k√100 = 10k, so k = 3.
- Step 3: At P = 400: dP/dt = 3√400 = 3(20) = 60.
Answer: dP/dt = 3√P; when P = 400, the population grows at 60 animals per year.
- Example 2
Trap: proportional to the difference
A cup of tea cools in a 70°F room. Its temperature T changes at a rate proportional to the difference between T and the room temperature. Write the differential equation. Is k positive or negative?
Show the solutionHide the solution
- Step 1: “The difference between T and room temperature” is T − 70. So dT/dt = k(T − 70).
- Step 2: The tea is hotter than the room, so T − 70 > 0, but the tea is cooling, so dT/dt < 0.
- Step 3: A positive number times k is negative, so k < 0.
- Step 4: Common wrong answer: dT/dt = kT. That would mean the tea cools toward 0°F, not toward the room's temperature.
Answer: dT/dt = k(T − 70), with k < 0.
Common mistakes
- Writing dy/dt = y + k instead of dy/dt = ky for “proportional to y.” Proportional means multiplied by a constant.
- Dropping the reference value in “proportional to the difference,” like writing kT instead of k(T − 70).
- Getting the sign of k wrong for something that is decreasing.
- Using the wrong letter for the independent variable when the problem uses names like P and t.
On the exam
- Multiple-choice questions often give a sentence and four differential equations. Look for the proportionality constant and check the sign.
- Free-response questions usually give the differential equation and ask about slopes, tangent lines or concavity at a point before asking you to solve it.
Connected topics
Videos
Check yourself
4 questions on 7.1 Modeling Situations with Differential Equations. Pick an answer to see if you got it, and why.
The amount A of a medication in a patient's bloodstream decreases at a rate proportional to the square root of the amount present. If t is time and k is a positive constant, which differential equation models this situation?
The rate of change of y with respect to t is inversely proportional to the square root of y. When y = 4, dy/dt = 3. What is dy/dt when y = 9?
A cup of tea is cooling in a room kept at a constant 20°C. The temperature T of the tea, in degrees Celsius, changes at a rate proportional to the difference between T and the room temperature. If t is time in minutes and k is a positive constant, which differential equation models the temperature of the tea?
A rumor spreads through a school of 1,200 students. The rate at which the number of students who have heard the rumor, N, changes with respect to time t is jointly proportional to the number of students who have heard the rumor and the number who have not. If k is a positive constant, which differential equation models this situation?
0 of 4 answered