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IB Maths AI HL · Unit 5: Calculus

IB Maths AI HL Differential Equations and Eulers Method Questions

Updated October 2026

Exam-style IB Maths AI HL differential equations and eulers method questions with worked solutions. Start with the 2 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.

52 questions
Paper 1: 38
Paper 2: 14
12 easy
20 medium
18 hard
2 very hard
2 worked examples

Practise Differential Equations and Eulers Method questions →

What you need to know

HL AI's Paper 3 favourite. Separable DEs by algebra, non-separable by Euler's numerical method. Practise the CG50 keystrokes for Euler. Differential equations — first-order separable and Euler's method overview →

HL AI extends SL AI with the chain, product and quotient rules and related rates. The 'water tank filling' family of questions turns on d/dt applied to a relationship between variables. Differentiation — chain, product and quotient rules, related rates overview →

Sketching solution curves from a slope field. HL AI Paper 3 loves 'without solving the DE, describe the long-term behaviour'. Slope fields and qualitative DE analysis overview →

What's examined in AI HL differential equations and eulers method

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

Solving First Order ODEs (26)
Systems and Phase Planes (15)
Euler's Method Approximation (11)

Key formulas

Separable differential equation: $\frac{d y}{d x} = f (x) g (y) ⟹ \int \frac{1}{g (y)} d y = \int f (x) d x$
Euler's method: $y_{n + 1} = y_{n} + h f (x_{n}, y_{n}), x_{n + 1} = x_{n} + h$
Trapezoidal rule (refresher): $A \approx \frac{h}{2} (y_{0} + y_{n} + 2 \sum_{i = 1}^{n - 1} y_{i})$

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

Where students lose marks on differential equations and eulers method

Examiner's noteEuler's method: formula copied, not applied, and GDC variables

Where marks are lost: Copying the booklet formula earns nothing; many wrote little except a final answer (full marks if right, none if wrong), used calculator notation, or wrote the iteration in the GDC's default variables. For systems and second-order equations the reports ask for the iteration in the question's variables, at least the first step shown, and the answer.

Examiners flagged this in 2 exam series (May 2024, May 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Differential Equations and Eulers Method worked examples

Worked example 1: Solving a Separable Differential Equation · easy

Use the method of separation of variables to solve the differential equation $\frac{d y}{d x} = \frac{3 x^{2}}{y^{2}}$ , given the initial condition $y (0) = 3$ . Express your answer in the form $y = f (x)$ .

Solution

6 steps · 1 to fill in

Examiner tip: Always add the constant of integration

+ C

immediately after performing the indefinite integration, before you start rearranging or isolating

y

Worked example 2: Applying Euler's Numerical Method · medium

Consider the differential equation $\frac{d y}{d x} = x - y^{2}$ with the initial condition $y (1) = 1$ . Use Euler's method with a step size of $h = 0.2$ to find an approximate value for $y (1.4)$ .

Solution

5 steps · 1 to fill in

Examiner tip: Setting up a table with columns for

n

x_{n}

y_{n}

, and

{(\frac{d y}{d x})}_{n}

is highly recommended to prevent substituting the wrong values into the iterative formula during exam pressure.

Try these IB Maths AI HL differential equations and eulers method 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 · 4 marks · Paper 1

Verify algebraically that the function $y = C e^{- 4 x}$ , where $C$ is an arbitrary constant, is a valid solution to the differential equation $\frac{d y}{d x} + 4 y = 0$ .

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 1

Find the exact particular solution to the differential equation: $\frac{d y}{d x} = y \cos x$ given the boundary condition that $y (0) = e^{2}$ .

Full worked solution Free, no sign-in

Separate variables: $\frac{1}{y} d y = \cos x d x$ M1
Integrate both sides: $\int \frac{1}{y} d y = \int \cos x d x$
$\ln | y | = \sin x + C$ A1 A1
Substitute $x = 0, y = e^{2}$ : $\ln (e^{2}) = \sin 0 + C$ M1
$2 = 0 + C ⟹ C = 2$ A1
Exact particular solution: $y = e^{\sin x + 2}$ (or $y = e^{2} e^{\sin x}$ ) A1

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

Question 3 · hard · 7 marks · Paper 1

A system of coupled differential equations is given by ${\vec{x}}^{'} = (\begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix}) \vec{x}$ .
The exact solution takes the form $\vec{x} (t) = c_{1} e^{λ_{1} t} {\vec{v}}_{1} + c_{2} e^{λ_{2} t} {\vec{v}}_{2}$ .
Find the exact solution for $\vec{x} (t)$ given the initial conditions $x (0) = 2$ and $y (0) = 1$ .

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI HL differential equations and eulers method questions are there?

There are 52 exam-style differential equations and eulers method questions in the AI HL question bank (Paper 1: 38 · Paper 2: 14), graded 12 easy, 20 medium, 18 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is differential equations and eulers method on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 38 · Paper 2: 14. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI 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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Free ready-to-teach lesson (slides and teacher notes): Separable first-order DEs.

IB Maths AI HL Differential Equations and Eulers Method Questions

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What you need to know

What's examined in AI HL differential equations and eulers method

Key formulas

Where students lose marks on differential equations and eulers method

Differential Equations and Eulers Method worked examples

Worked example 1: Solving a Separable Differential Equation · easy

Solution

Worked example 2: Applying Euler's Numerical Method · medium

Solution

Try these IB Maths AI HL differential equations and eulers method questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise differential equations and eulers method for the exam

FAQ

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