Differential Equations

Name: AP Calculus BC Unit 7: Differential Equations
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6–9% of the exam4 lessons · 56 min7 terms

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 7. This unit is worth 6–9% of the exam, so budget your time against that rather than against how long the unit takes to teach.

LogisticEuler’s methodSlope fieldsSeparation

Lessons in this unit

Slope Fields13 min · 3 objectivesInterpret a slope field as a picture of a differential equation · Sketch a solution curve through a given point on a slope field · Match a differential equation to its slope field
Separation of Variables15 min · 3 objectivesSolve separable differential equations by separating and integrating · Apply an initial condition to find the particular solution · Recognize exponential growth and decay as separable models
Euler’s Method14 min · 3 objectivesApproximate a solution numerically using Euler’s method · Carry out repeated steps with a fixed step size · Explain why Euler’s method produces an approximation, not an exact value
The Logistic Model14 min · 3 objectivesInterpret the logistic differential equation and its carrying capacity · Identify the population level of fastest growth · Determine the long-run limit of a logistic solution

Formulas in Unit 7

What a slope field encodes

At each point (x, y), segment slope = dy/dx = f(x, y)

The field is the geometry of the equation; a solution curve threads through it tangent to every segment.

Exponential growth/decay model

dy/dt = k·y ⟹ y = y₀·e^{kt}

The classic separable equation: rate proportional to amount. k > 0 is growth, k < 0 is decay; y₀ is the initial amount.

Euler’s method update

x_{n+1} = x_n + Δx · y_{n+1} = y_n + f(x_n, y_n)·Δx

The new y is the old y plus slope times step size. "Rise = slope × run" applied one step at a time.

Logistic differential equation

dP/dt = kP(1 − P/M)

M is the carrying capacity. Growth rate is near-exponential for small P and falls to 0 as P → M.

Every term in Unit 7

All 7 terms we publish for Differential Equations, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Logistic differential equation: dP/dt = kP(1 − P/L). Growth is nearly exponential when P is small and slows to zero as P approaches the carrying capacity L.
Euler's method: Step forward along the tangent: y_{n+1} = y_n + h·f(x_n, y_n). It underestimates a concave-up solution because tangent lines lie below the curve.
Choosing the step size: Smaller h gives a better approximation and more arithmetic. Halving h roughly halves the error for this first-order method.
Fastest growth in a logistic model: Occurs at P = L/2, which is the inflection point of the solution curve. A standard BC free-response question.
Logistic solution behavior: As t→∞, P approaches L from whichever side it started, so L is a horizontal asymptote of the solution.
Identifying the carrying capacity: Set dP/dt = 0 and solve; the non-zero equilibrium is L. It can be read directly from the logistic form.
Separation of variables with initial conditions: Integrate both sides, add one constant, apply the initial condition immediately, then solve for the dependent variable.

What examiners penalize here

When a free-response question shows a slope field, you can sketch a particular solution without solving the equation — just start at the given point and follow the segments. Full credit comes from a curve tangent to the field, passing through the initial condition.
On free response you must show the separation and the integration of both sides, then apply the initial condition explicitly. A bare final formula, even if correct, typically earns only partial credit — the graders reward the separated equation and the antiderivatives.
Organize Euler’s method in a small table with columns for x, y, slope f(x,y), and Δy = slope·Δx. The table structure prevents arithmetic slips across steps and is exactly what free-response graders expect to see.
You can often answer logistic questions without solving the differential equation. Read M directly from the equation as the P that zeroes the (1 − P/M) factor, then use the two facts: fastest growth at M/2, and the long-run limit is M.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

42 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Calculus BC study guideAll units, weights and score data in one place All 10 Calculus BC unitsThe full course

Questions about this unit

How much of the AP Calculus BC exam is Unit 7?

Unit 7, Differential Equations, is worth 6–9% of the Calculus BC multiple-choice section according to the published course framework. Across all 10 units that makes it a middling share, roughly what an even split across units would give.

What topics are covered in Calculus BC Unit 7?

Differential Equations covers Logistic, Euler’s method, Slope fields and Separation. We publish 7 terms with definitions for this unit, all of them on this page.

How should I study Calculus BC Unit 7?

Read the 4 lessons below first — about 55 minutes — then drill the 7 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 10 units of AP Calculus BC

Unit names, topics and exam weights follow the published College Board course framework for AP Calculus BC. AP® is a trademark registered by the College Board, which does not endorse this site.