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IB Maths AA HL · Unit 5: Calculus

IB Maths AA HL Differential Equations and Maclaurin Questions

Updated October 2026

Exam-style IB Maths AA HL differential equations and Maclaurin questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

62 questions
Paper 1: 42
Paper 2: 20
12 easy
31 medium
19 hard
3 worked examples

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What you need to know

HL AA covers first-order separable DEs and linear DEs solved by integrating factor. Paper 3 often layers a Maclaurin series check on top. Differential equations — separable and integrating factor overview →

HL AA extends SL AA's toolkit with implicit differentiation, logarithmic differentiation (for y = x^x-style functions), and derivatives of arcsin, arccos, arctan. Differentiation — implicit, logarithmic, and inverse-trig overview →

HL AA Paper 3 favourite. Derive the Maclaurin series for e^x, sin(x), cos(x), ln(1+x), and use them for approximations plus limit calculations. Maclaurin series and approximations overview →

What's examined in AA HL differential equations and Maclaurin

The question bank covers these differential equations and Maclaurin question types (number of questions in brackets):

Solving Differential Equations (28)
Maclaurin Series (19)
Euler's Method (15)

Key formulas

Euler's identity: $e^{i π} + 1 = 0$
Derivative of tan x: $\frac{d}{d x} (\tan x) = \sec^{2} x$
Maclaurin series: $f (x) = f (0) + f^{'} (0) x + \frac{f^{″} (0)}{2!} x^{2} + \frac{f^{‴} (0)}{3!} x^{3} + \dots$
Standard Maclaurin expansions: $e^{x} = \sum_{k = 0}^{\infty} \frac{x^{k}}{k!}, \sin x = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} x^{2 k + 1}}{(2 k + 1)!}, \cos x = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} x^{2 k}}{(2 k)!}$

In the same notation as the IB formula booklet. All AA HL formulas →

Differential Equations and Maclaurin worked examples

Worked example 1: Maclaurin series using standard expansions · easy

Find the first three non-zero terms of the Maclaurin series expansion for $f (x) = x \cos (2 x)$ .

Solution

6 steps

Examiner tip: Substituting composite arguments into the standard Maclaurin series provided in the formula booklet is vastly quicker and less error-prone than repeatedly applying the product rule to differentiate the function from scratch.

Worked example 2: Separable differential equations · medium

Solve the differential equation $\frac{d y}{d x} = \frac{x^{2}}{y}$ , given the boundary condition that $y (0) = 2$ . Give your answer in the explicitly solved form $y = f (x)$ .

Solution

6 steps

Examiner tip: Do not forget the constant of integration

+ C

; it must be added immediately after integrating, before any algebraic rearrangement or square roots. Putting it at the end of the calculation will completely ruin the solution.

Worked example 3: First-order DEs with an integrating factor · hard

Find the general solution to the first-order linear differential equation $\frac{d y}{d x} + 3 y = e^{2 x}$ . Give your answer in the form $y = f (x)$ .

Solution

6 steps

Examiner 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

\frac{d y}{d x}

term before identifying

P (x)

Try these IB Maths AA HL differential equations and Maclaurin questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 3 marks · Paper 1

Consider the linear differential equation: $\frac{d y}{d x} + \frac{2}{x} y = 5 x$ Find the integrating factor required to solve this differential equation.

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Question 2 · medium · 6 marks · Paper 2

According to Newton’s Law of Cooling, the rate at which an object cools is proportional to the difference between its temperature $T$ and the ambient room temperature $T_{A}$ . This is modelled by $\frac{d T}{d t} = - k (T - T_{A})$ .
A cup of coffee is initially $100^{\circ} C$ in a room kept at a constant $20^{\circ} C$ . After 10 minutes, the coffee has cooled to $60^{\circ} C$ .
Calculate the exact temperature of the coffee after 20 minutes.

Full worked solution Free, no sign-in

With $T_{A} = 20$ , separate the variables:
$\int \frac{1}{T - 20} d T = - \int k d t$ M1
$\ln (T - 20) = - k t + c$ , so $T = 20 + A e^{- k t}$ A1
Use $T = 100$ at $t = 0$ :
$A = 80$ A1
Use $T = 60$ at $t = 10$ :
$40 = 80 e^{- 10 k} ⟹ e^{- 10 k} = \frac{1}{2}$ M1A1
At $t = 20$ , $e^{- 20 k} = {(e^{- 10 k})}^{2} = \frac{1}{4}$ :
$T = 20 + 80 \cdot \frac{1}{4} = 40^{\circ} C$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 1

A function $y = f (x)$ satisfies the differential equation: $(1 - x) \frac{d y}{d x} = 2 y$ Given that $y (0) = 1$ , find the Maclaurin series expansion for $y$ up to and including the term in $x^{3}$ by repeatedly differentiating the differential equation.

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FAQ

How many IB Maths AA HL differential equations and Maclaurin questions are there?

There are 62 exam-style differential equations and Maclaurin questions in the AA HL question bank (Paper 1: 42 · Paper 2: 20), graded 12 easy, 31 medium, 19 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is differential equations and Maclaurin on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 42 · Paper 2: 20. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA HL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Separable first-order DEs · Integrating-factor DEs · Maclaurin series · Euler's method for differential equations.

IB Maths AA HL Differential Equations and Maclaurin Questions

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What you need to know

What's examined in AA HL differential equations and Maclaurin

Key formulas

Differential Equations and Maclaurin worked examples

Worked example 1: Maclaurin series using standard expansions · easy

Solution

Worked example 2: Separable differential equations · medium

Solution

Worked example 3: First-order DEs with an integrating factor · hard

Solution

Try these IB Maths AA HL differential equations and Maclaurin questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 6 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Revise differential equations and Maclaurin for the exam

FAQ

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