STEM-TutorLearn • Practice • ExcelBook Free DemoFree demo
AP CoursesGCSEInstructorsTestimonials
Guides
Past papers
ContactBook Free Demo

Differential equations practice problems AP Calculus, Unit 7

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Slope fields, separable differential equations, exponential models, and (for BC) Euler's method and logistic growth. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for differential equations

dy/dx = ky ⇒ y = y0ekxexponential growth or decay
Separate: ∫ dy/g(y) = ∫ f(x) dxseparable equations
ynew = y + h × (dy/dx)Euler's method (BC)

Questions marked No calculator should be done by hand, as in Part A of the multiple-choice section. You can type fractions, such as 3/4. The full sheet is in our AP Calculus formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-upNo calculator
Which function is a solution of dy/dx = 2y?

Hint

Substitute each one: is its derivative twice itself?

Worked solution
  1. For y = e2x, dy/dx = 2e2x = 2y.
  2. None of the others satisfies the equation. The answer is C.
Q2Warm-upNo calculator
In the slope field for dy/dx = x − y, what is the slope at (2, 1)?

Hint

Substitute the point into dy/dx.

Worked solution
  1. dy/dx = 2 − 1 = 1.
Q3Warm-upNo calculator
Solve dy/dx = 6x2 with y(1) = 3, and find y(2).

Hint

Integrate, then use the initial condition to find C.

Worked solution
  1. y = 2x3 + C.
  2. y(1) = 2 + C = 3, so C = 1.
  3. y(2) = 16 + 1 = 17.

Exam-style questions

Questions at the level of the AP Calculus exam. The ones marked Challenge combine more than one idea.

Q4Exam-styleCalculator
Solve dy/dx = xy with y(0) = 2. Find y(1), to 3 decimal places.

Hint

Separate the variables: dy/y = x dx.

Worked solution
  1. ln|y| = x2/2 + C, so y = Aex2/2.
  2. y(0) = 2 gives A = 2.
  3. y(1) = 2e0.5 = 3.297.
Q5Exam-styleCalculator
A population satisfies dP/dt = 0.05P with P(0) = 1000. Find P(10), to the nearest whole number.

Hint

This is exponential growth.

Worked solution
  1. P(t) = 1000e0.05t.
  2. P(10) = 1000e0.5.
  3. ≈ 1649.
Q6Exam-styleBC only
Use Euler's method with step 0.5 for dy/dx = x + y, y(0) = 1, to estimate y(1).

Hint

Two steps, each: new y = y + 0.5 × (x + y).

Worked solution
  1. Step 1: y(0.5) ≈ 1 + 0.5(0 + 1) = 1.5.
  2. Step 2: y(1) ≈ 1.5 + 0.5(0.5 + 1.5) = 2.5.
Q7Exam-styleNo calculator
dy/dt = ky with k < 0 models:

Hint

The solution is y = y0ekt. What happens when k is negative?

Worked solution
  1. y = y0ekt with k < 0 shrinks by the same factor each unit of time.
  2. That is exponential decay: B.
Q8ChallengeCalculator
Solve dy/dx = (y − 1)/x2 with y(1) = 3. Find y(2), to 3 decimal places.

Hint

Separate: dy/(y − 1) = dx/x2.

Worked solution
  1. ln|y − 1| = −1/x + C. At (1, 3): ln 2 = −1 + C, so C = 1 + ln 2.
  2. y − 1 = 2e1 − 1/x.
  3. y(2) = 1 + 2e0.5 = 4.297.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

No-calculator free-response style · 6 points · about 15 minutes

Consider the differential equation dy/dx = (y − 1)/x2, with y(1) = 3.

  1. (a) Write the equation of the tangent line to the solution at x = 1, and use it to estimate y(1.1).
  2. (b) Find the particular solution y = f(x).

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

Slope at (1, 3) = 2. y ≈ 3 + 2(x − 1), so y(1.1) ≈ 3.2.

Part (b)

dy/(y − 1) = dx/x2; ln|y − 1| = −1/x + C; C = 1 + ln 2; y = 1 + 2e1 − 1/x.

Your mark: 0 of 6

Common mistakes in differential equations

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Calculus past papers, with official scoring guidelines. · All AP Calculus units