Review AP Calculus AB/BC Unit 7 to build fluency with differential equations, from modeling and slope fields to separation of variables and exponential or logistic growth models. This unit ties together integration skills from Unit 6 with real-world rate-of-change contexts that appear consistently on both the AB and BC exams.
Use the topic guides, practice questions, FRQ practice, and AP score calculator available on Fiveable to work through every concept in this unit.
A differential equation is any equation that relates a function to one or more of its derivatives. Unit 7 asks you to work with these equations in four main ways: modeling a situation by writing the equation, verifying that a proposed function is a solution, estimating solutions graphically or numerically, and solving analytically using separation of variables.
Topics 7.1 and 7.2 focus on translating language like 'the rate of change is proportional to the quantity' into dy/dt = ky, and then checking whether a given function satisfies the equation by differentiating and substituting back in.
Topics 7.3 and 7.4 use slope fields to visualize solution families without solving the equation. Topic 7.5 (BC only) extends this numerically with Euler's method: y_{n+1} = y_n + h f(x_n, y_n).
Topics 7.6 through 7.9 cover the full solution process: separate variables, integrate both sides, apply an initial condition to find a particular solution, and interpret exponential (dy/dt = ky) or logistic (dy/dt = ky(a - y)) models in context.
Every differential equation in this unit says something about how fast a quantity is changing. The solution is the function that produces exactly that rate of change. Separation of variables works by treating dy/dx as a ratio, moving all y terms to one side and all x terms to the other, then integrating. The constant of integration C produces a family of general solutions; one initial condition pins down the unique particular solution.
Translate verbal descriptions of rates of change into differential equations. Key patterns include proportional growth (dy/dt = ky), Newton's law of cooling, and mixing problems.
Differentiate a proposed function, substitute it and its derivative into the ODE, and confirm both sides match. Understand the difference between general solutions (with C) and particular solutions.
Evaluate dy/dx = f(x, y) at grid points and draw short segments with those slopes. Recognize equilibrium solutions as horizontal rows of zero-slope segments.
Trace solution curves through a slope field using an initial condition. Identify stable and unstable equilibria and describe long-term behavior from the field's visual pattern.
Apply the update rule y_{n+1} = y_n + h * f(x_n, y_n) iteratively from an initial condition. Smaller step sizes improve accuracy; concavity determines whether the method over- or underestimates.
Rewrite dy/dx = f(x)g(y) as dy/g(y) = f(x) dx, integrate both sides, and include the constant C. The result is a general solution representing a family of curves.
After finding the general solution, substitute the initial condition to solve for C. Check domain restrictions and recognize the definite-integral form F(x) = y_0 + integral from a to x of f(t) dt.
The model dy/dt = ky has solution y = y_0 * e^(kt). Interpret k and y_0 in context, compute half-life or doubling time, and connect the model to separation of variables.
Interpret dy/dt = ky(a - y) without solving: carrying capacity is a, fastest growth is at y = a/2, and the long-term limit is a. Recognize the S-shaped solution curve.
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Review Logistic Models with Differential Equations (BC Only) with attention to how the concept appears in AP-style source and evidence questions.
Review Approximating Solutions Using Euler's Method (BC Only) with attention to how the concept appears in AP-style source and evidence questions.
Review Exponential Models with Differential Equations with attention to how the concept appears in AP-style source and evidence questions.
Review Finding Particular Solutions Using Initial Conditions and Separation of Variables with attention to how the concept appears in AP-style source and evidence questions.
A differential equation connects a function to its derivative. The key skill in 7.1 is reading a verbal description and writing the correct equation. Watch for proportionality language: 'the rate of change of y is proportional to y' becomes dy/dt = ky. Other common setups include Newton's law of cooling, dT/dt = -k(T - T_env), and mixing problems where dA/dt = rate in minus rate out.
| Verbal description | Differential equation |
|---|---|
| Rate of change proportional to quantity | dy/dt = ky |
| Rate of change proportional to difference from ambient | dT/dt = -k(T - T_env) |
| Rate of change jointly proportional to y and (a - y) | dy/dt = ky(a - y) |
To verify that a function is a solution, differentiate it, substitute both the function and its derivative into the differential equation, and confirm both sides are equal for all x. A general solution contains an arbitrary constant C and represents infinitely many solution curves. A particular solution has C determined by an initial condition.
| Solution type | Contains C? | Satisfies initial condition? |
|---|---|---|
| General solution | Yes | No (C is free) |
| Particular solution | No (C is fixed) | Yes |
A slope field draws a short line segment at each grid point (x, y) whose slope equals dy/dx evaluated there. You read the field by tracing a path that always stays tangent to the local segments. An initial condition tells you which solution curve to follow. Equilibrium solutions appear as horizontal segments across an entire row; stable equilibria attract nearby curves, unstable ones repel them.
| Feature | Autonomous ODE | Non-autonomous ODE |
|---|---|---|
| Slope depends on | y only | Both x and y |
| Horizontal rows | Identical slopes | Slopes vary across row |
| Equilibrium analysis | Phase line works | Phase line not sufficient |
Euler's method approximates a solution curve by taking small linear steps. Starting from the initial condition (x_0, y_0), compute the slope f(x_0, y_0), step forward by h to get x_1 = x_0 + h and y_1 = y_0 + h * f(x_0, y_0), then repeat. Smaller step sizes give better approximations. Organize each iteration in a table showing x_n, y_n, f(x_n, y_n), and y_{n+1}.
| Concavity of true solution | Euler approximation error direction |
|---|---|
| Concave up | Underestimate |
| Concave down | Overestimate |
Separation of variables applies when dy/dx = f(x) * g(y). Rewrite as dy/g(y) = f(x) dx, integrate both sides, and include a constant of integration C on one side. The result is a general solution. To find a particular solution, substitute the initial condition (x_0, y_0) and solve for C. Watch for domain restrictions: if you divided by g(y), check whether g(y) = 0 produces additional equilibrium solutions.
| Step | What you do | Common error |
|---|---|---|
| Separate | Move all y terms left, all x terms right | Forgetting to divide by g(y) |
| Integrate | Integrate both sides | Omitting the constant C |
| Apply IC | Substitute (x_0, y_0) and solve for C | Applying IC before integrating |
| Check domain | Identify where solution is valid | Ignoring log or square-root restrictions |
When a quantity's rate of change is proportional to its current size, the model is dy/dt = ky. Solving by separation of variables gives y = y_0 * e^(kt). If k > 0 the quantity grows; if k < 0 it decays. Common applications include population growth, radioactive decay, and continuously compounded interest. The sign and magnitude of k carry meaning in context: always interpret them with units.
| Feature | Exponential growth (k > 0) | Exponential decay (k < 0) |
|---|---|---|
| Long-term behavior | Grows without bound | Approaches zero |
| Graph shape | Increasing, concave up | Decreasing, concave up |
| Key time measure | Doubling time ln(2)/k | Half-life ln(2)/|k| |
The logistic differential equation dy/dt = ky(a - y) models growth that accelerates early and then slows as y approaches the carrying capacity a. You do not need to solve the equation to answer most exam questions: the carrying capacity is a, the fastest growth occurs when y = a/2, and the solution has a horizontal asymptote at y = a. The equilibrium solutions are y = 0 (unstable) and y = a (stable).
| Feature | Exponential model dy/dt = ky | Logistic model dy/dt = ky(a - y) |
|---|---|---|
| Long-term behavior | Grows without bound (k > 0) | Levels off at carrying capacity a |
| Fastest growth | Always accelerating | At y = a/2 |
| Equilibria | y = 0 only | y = 0 (unstable) and y = a (stable) |
| Curve shape | Pure exponential | S-shaped sigmoid |
Try AP-style multiple-choice questions and written prompts after you review the notes.
| Term | Definition |
|---|---|
| dy/dt = ky | The differential equation model for exponential growth and decay; the rate of change of a quantity is proportional to its current size, with k as the constant of proportionality. |
| General Solution | A family of functions satisfying a differential equation, containing an arbitrary constant C that represents infinitely many solution curves. |
| Particular Solution | The unique solution to a differential equation obtained by using an initial condition to determine the value of the constant C. |
| Initial Condition | A specified value y(x_0) = y_0 that identifies one point on a solution curve and determines the particular solution. |
| Constant of Integration | The arbitrary constant C added after integrating during separation of variables; it must be included before applying any initial condition. |
| Slope Field | A grid of short line segments where each segment at (x, y) has slope equal to dy/dx evaluated at that point, giving a visual picture of all solution curves. |
| Solution Curve | A curve drawn through a slope field that is tangent to every segment it passes through; each initial condition produces a different solution curve. |
| Euler's Method | A numerical technique (BC only) that approximates a solution curve using the update rule y_{n+1} = y_n + h * f(x_n, y_n), stepping forward from an initial condition with step size h. |
| Rate of Change | The derivative of a function; in differential equations, it describes how quickly a quantity is changing and is the central quantity being modeled. |
| Exponential Decay | The behavior of y = y_0 * e^(kt) when k < 0; the quantity decreases toward zero and has a half-life of ln(2) / |k|. |
| Newton's Law of Cooling | The differential equation dT/dt = -k(T - T_env), which models a temperature T approaching the ambient temperature T_env at a rate proportional to their difference. |
| Domain | The interval on which a particular solution is valid; separation of variables can introduce logarithms or fractions that restrict where the solution exists. |
| Tangent Line | In the context of slope fields and Euler's method, the line whose slope equals dy/dx at a given point; it is the basis for each linear step in Euler's method. |
After integrating both sides during separation of variables, students often forget to write + C. The constant must appear before you apply any initial condition; leaving it out means you are solving for a particular solution without using the initial condition properly.
Substituting the initial condition into the unseparated or partially separated equation instead of into the integrated general solution is a common procedural error. Always integrate first, then solve for C.
When sketching a solution curve through a slope field, students sometimes draw curves that cross segments at the wrong angle or ignore the local slope entirely. The curve must be tangent to every segment it passes through.
The carrying capacity is a in dy/dt = ky(a - y), but the fastest growth occurs at y = a/2, not at y = a. These are two different values and the exam often asks about both.
If a quantity is decaying, k must be negative in dy/dt = ky and y = y_0 * e^(kt). Writing k as a positive number and then subtracting it separately, or confusing the sign when computing half-life, leads to incorrect answers.
Free-response questions in this unit frequently ask you to write a differential equation from a verbal description, solve it by separation of variables, apply an initial condition, and then interpret the result in context. Showing each algebraic step clearly, including the constant of integration and the domain of the solution, is expected for full credit.
Multiple-choice and free-response items may show a slope field and ask you to identify which differential equation it represents, sketch a particular solution through a given point, or describe the long-term behavior of solutions. These tasks require you to evaluate dy/dx at specific points and trace curves that stay tangent to the field.
Both AB and BC exams include questions where you must identify the model type from an equation or description, extract key quantities such as the growth constant, carrying capacity, or time of fastest growth, and connect the differential equation to its solution form. BC questions on logistic growth often ask you to reason about behavior directly from dy/dt = ky(a - y) without solving the equation.
Open the individual guides for Unit 7 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Calc Unit 7 covers 9 topics across differential equations: modeling situations with differential equations, verifying solutions, sketching and reasoning with slope fields, separation of variables for general and particular solutions, exponential models, and (BC only) Euler's method and logistic models. See AP Calc Unit 7 for matched practice on each topic.
Unit 7 makes up 6-12% of the AP Calc exam. That weight covers everything from sketching slope fields and solving separable differential equations to modeling exponential growth and decay. It's a focused unit, but the FRQ section often pulls directly from separation of variables and initial condition problems, so the payoff for studying it is high.
The AP Calc Unit 7 progress check includes both MCQ and FRQ parts drawn from this unit's core topics. The MCQ section tests slope field reasoning, verifying solutions, and setting up separable differential equations. The FRQ part typically asks you to find a general or particular solution using separation of variables and an initial condition, and may include an exponential or logistic model (BC). Use AP Calc Unit 7 to find practice that mirrors the progress check format.
AP Calc Unit 7 FRQs most often ask you to solve a separable differential equation, apply an initial condition to find a particular solution, or interpret a slope field. To practice, work through problems that start with a given dy/dx expression, separate variables, integrate both sides, and solve for the constant using a point on the curve. Exponential growth and decay setups are especially common. Find FRQ-style problems at AP Calc Unit 7 to build that step-by-step fluency.
For AP Calc Unit 7 practice questions, including MCQ and practice test problems, head to AP Calc Unit 7. There you'll find multiple-choice questions on slope fields, verifying solutions, and separation of variables, plus free-response practice covering particular solutions and exponential models. Mixing MCQ and FRQ practice is the best way to prepare for how this unit shows up on the full exam.
Start with slope fields (7.3 and 7.4) since they build intuition for what a differential equation is actually showing you. Then work through separation of variables (7.6) until the algebra feels automatic, and move to particular solutions with initial conditions (7.7). From there, exponential models (7.8) will click quickly. BC students should add Euler's method (7.5) and logistic models (7.9) last. Practice by writing out every integration step, checking your constant of integration, and sketching the solution curve. AP Calc Unit 7 has topic-by-topic resources to work through in that order.