Free IB Maths AA HL lesson: Integrating-factor DEs
A ready-to-teach IB Maths AA HL lesson on Integrating-factor DEs (Unit 5 · Calculus): editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
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What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans4 questions, 17 marks: Fluency, Reasoning, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- ∫ ln x dx & Cyclic Integration by Parts
- The Integrating Factor Method
Key ideas on the slides
- Write the equation as dy/dx + P(x)y = Q(x).
- The integrating factor is I(x) = e^(∫ P(x) dx).
- Multiply through: (d/dx)(I(x)y) = I(x)Q(x), then integrate: I(x)y = ∫ I(x)Q(x) dx (add the constant here).
- Example: dy/dx + (2/x)y = x has I = x², so x²y = x⁴/4 + C.
By the end of the lesson, students should be able to answer a question like: Solve dy/dx + 2y = e^x using an integrating factor.
Key vocabulary
- integrating factor: e^(∫ P dx), which makes the left side an exact derivative.
- linear differential equation: A first-order DE of the form dy/dx + P(x)y = Q(x).
- standard form: The coefficient of dy/dx is 1.
Common mistakes this lesson tackles
- Differential equations: integrating factor and the constant of integration. For dy/dx + P(x)y = Q(x): integrating factor e^(∫P dx), then (IF × y)′ = IF × Q. Add + C at the integration step and use the condition to find C before rearranging.
- Not dividing through to make the coefficient of dy/dx equal to 1.
- Forgetting to multiply the right-hand side by the integrating factor.
- Adding the constant after dividing by I(x).
The worked example
Find the general solution to the first-order linear differential equation dy/dx + 3y = e²ˣ. Give your answer in the form y = f(x).
- Recognize the standard linear form dy/dx + P(x)y = Q(x), where P(x) = 3 and Q(x) = e²ˣ.
- Calculate the integrating factor (IF) using e^(∫ P(x) dx): IF = e^(∫ 3 dx) = e³ˣ.
- Multiply the entire differential equation by the integrating factor: e³ˣ dy/dx + 3e³ˣ y = e³ˣ e²ˣ.
- Condense the left-hand side using the reverse product rule: (d/dx)(y e³ˣ) = e⁵ˣ.
- Integrate both sides with respect to x: y e³ˣ = ∫ e⁵ˣ dx ⇒ y e³ˣ = ⅕ e⁵ˣ + C.
- Isolate y by dividing all terms by e³ˣ: y = ⅕ e²ˣ + C e⁻³ˣ.
Tip: When solving by an integrating factor, always ensure the differential equation is arranged perfectly into the standard form with a leading coefficient of 1 on the dy/dx term before identifying P(x).
Calculator tips
- Use a numerical DE solver (if available) to check your particular solution at one point.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Separable first-order DEs and Kinematics: displacement, velocity, acceleration |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (4 questions in 3 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens the IB Maths AA HL Unit 5: Differential Equations and Maclaurin practice questions.
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (with a School plan).
- This lesson's worksheet in three levels, with its answers, ready to print.
- Support, Core and Stretch versions and A/B copies of the worksheet from the full question bank, in the worksheet builder.
- Create lesson: this lesson rebuilt for your class's next scheme-of-work lesson, from the full bank.
School licences from €490 a year. Try everything free for 30 days as a teacher: no card.
Questions teachers ask
Is this lesson free?
The slides and the teacher notes are free to download with a free IB Math Revision teacher account (sign in with Google, Microsoft or email); no card. The printable worksheet in three levels and its answers are included with school plans and the free 30-day teacher trial.
Can I edit the slides?
Yes. The slides are an ordinary PowerPoint file: change, add or delete anything. They also open in Google Slides and Keynote. The logo and web address sit on the slide master, so your edits do not move them.
What do students use for homework?
The practice slide's QR code and web address take students to the IB Maths AA HL Unit 5: Differential Equations and Maclaurin practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.