Modeling Situations with Differential Equations
Translate a rate statement into a differential model with variables, signs, units, and an initial condition.
College Board Topic 7.1 · CED p. 132 (PDF p. 137) · CED effective Fall 2020; project course outline 2026–27.
A differential equation connects a quantity to its rate of change. Translate the words first: “the rate is proportional to the amount” means y′=ky, while “the rate is proportional to the difference from 20” means y′=k(20−y). Define the independent variable, the dependent quantity, and the units of every constant. An initial condition such as y(0)=5 gives a starting state; it is not another derivative.
Motion models can say s′(t)=v(t) or v′(t)=a(t). A tank with inflow I(t) and outflow O(t) obeys V′=I−O. If a temperature moves toward a room temperature of 20°C, T′=−k(T−20) with k>0 has the right direction: hotter objects cool and colder objects warm. A model needs a plausible interval and units, and its solution should respect any physical restrictions.
Let's work through it
- Let V(t) be liters after t minutes. Inflow is 5 L/min and outflow is 0.1V L/min. Reason: Defining variables makes the rate statement precise.
- Write V′(t)=5−0.1V(t), with 0.1 measured in min⁻¹. Reason: Net volume rate is inflow minus outflow.
- If V(0)=20 L, the starting rate is 5−0.1(20)=3 L/min. Reason: Substitute the initial state into the model.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Does y′=3y mean y increases by 3 units each minute?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Does y′=3y mean y increases by 3 units each minute?
Worked answer and reason
No. Its instantaneous rate is three times its current amount.
A proportional rate depends on the current state.
Retrieval 2
For position s, what does s″ represent?
Worked answer and reason
Acceleration.
Velocity is the first derivative of position.
Catch the tempting mistake
- The initial amount belongs in the derivative equation as an added rate.
- An initial amount is a separate condition. Adding it to a rate mixes quantities with different units.
Make the picture move
AB: slope fields · BC: Euler’s method and logistic models
Use the slope now to take the next step.
Try first. Predict whether forward Euler steps underestimate or overestimate this increasing, concave-up solution. Then halve the step size.
Scroll sideways to see the full graph on a small screen. The table below gives the numerical values.
- Exact solution
- Forward Euler path
With 8 steps of size 0.25, Euler gives y(2) ≈ 5.9605; the exact value is 7.3891.
Compare the numbers
| Time t | Euler estimate | Exact solution | Absolute error |
|---|---|---|---|
| 0 | 1 | 1 | 0 |
| 0.5 | 1.5625 | 1.6487 | 0.0862 |
| 1 | 2.4414 | 2.7183 | 0.2769 |
| 1.5 | 3.8147 | 4.4817 | 0.667 |
| 2 | 5.9605 | 7.3891 | 1.4286 |
What’s going on? Euler uses y(next) = y + Δt · y. Each step follows the slope at the start of that step. The exact solution is e^t; because it bends upward, these forward tangent steps lie below it.
Keep in mind. Euler’s method and logistic models are BC topics. The displayed exact solution is a comparison, not the output of Euler’s method. Numerical error depends on step size and the differential equation; these safe controls do not establish a general stability result.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 7 problem 1 · No calculator · verbal
- Unit 7 problem 2 · No calculator · verbal
- Unit 7 problem 3 · No calculator · verbal
For the teacher
- Have students mark noun, rate phrase, sign, and units in a verbal problem. Require them to test the equation above and below an equilibrium value.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Initial condition
An initial condition specifies a solution’s value at a particular input. It can determine the constant in a family of solutions to a differential equation. For example, y′=2x with y(0)=3 selects y=x²+3. The solution must satisfy both the equation and the stated condition on a suitable interval.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.