IB Maths AI HL · Unit 5: Calculus
IB Maths AI HL Differential Equations and Eulers Method Questions
Exam-style IB Maths AI HL differential equations and eulers method questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 19 questions
- Paper 1: 19
- 4 easy
- 4 medium
- 4 hard
- 7 starter
- 3 worked examples
Practise Differential Equations and Eulers Method questions →
AI HL formula booklet
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):
- Systems and Phase Planes (9)
- Solving First Order ODEs (7)
- Euler's Method Approximation (3)
Key formulas
- Separable differential equation
- \(\dfrac{dy}{dx} = f(x) g(y) \implies \int \dfrac{1}{g(y)}\, dy = \int f(x)\, dx\)
- Euler's method
- \(y_{n+1} = y_n + h\, f(x_n, y_n),\ x_{n+1} = x_n + h\)
- Trapezoidal rule (refresher)
- \(A \approx \tfrac{h}{2}\bigl(y_0 + y_n + 2\sum_{i=1}^{n-1} y_i\bigr)\)
In the same notation as the IB formula booklet. All AI HL formulas →
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 $\frac{dy}{dx} = \frac{3x^2}{y^2}$, given the initial condition $y(0) = 3$. Express your answer in the form $y = f(x)$.
1. Separate the variables by moving all $y$ terms to the left and $x$ terms to the right: $y^2 \, dy = 3x^2 \, dx$.
2. Integrate both sides: $\int y^2 \, dy = \int 3x^2 \, dx$.
3. Evaluate the integrals: $\frac{y^3}{3} = x^3 + C$.
4. Substitute the initial condition $x=0, y=3$ to find the constant $C$: $\frac{3^3}{3} = 0^3 + C \implies \frac{27}{3} = C \implies C = 9$.
5. Substitute $C$ back into the equation: $\frac{y^3}{3} = x^3 + 9 \implies y^3 = 3x^3 + 27$.
6. Isolate $y$ by taking the cube root to find the explicit exact solution: $y = \sqrt[3]{3x^3 + 27}$.
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 $\frac{dy}{dx} = x - y^2$ 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 $y(1.4)$.
1. Identify the initial values for Step 1: $x_0 = 1$, $y_0 = 1$. Calculate the initial gradient: $\left(\frac{dy}{dx}\right)_0 = 1 - 1^2 = 0$.
2. Apply Euler's formula $y_{n+1} = y_n + h \left(\frac{dy}{dx}\right)_n$ to find $y_1$: $y_1 = 1 + 0.2(0) = 1$.
3. Identify the values for Step 2: $x_1 = 1 + 0.2 = 1.2$, $y_1 = 1$. Calculate the new gradient: $\left(\frac{dy}{dx}\right)_1 = 1.2 - 1^2 = 0.2$.
4. Apply Euler's formula to find $y_2$ (which corresponds to $x=1.4$): $y_2 = 1 + 0.2(0.2)$.
5. Calculate the final approximation: $y_2 = 1 + 0.04 =$ $1.04$.
Examiner tip: Setting up a table with columns for $n$, $x_n$, $y_n$, and $\left(\frac{dy}{dx}\right)_n$ is highly recommended to prevent substituting the wrong values into the iterative formula during exam pressure.
Worked example 3: Using an Integrating Factor · hard
A differential equation is given in standard linear form as $\frac{dy}{dx} + \frac{2}{x} y = 4x$ for $x > 0$. Solve the differential equation to find the general solution for $y$ in terms of $x$.
1. Identify $P(x) = \frac{2}{x}$ and formulate the integrating factor: $I(x) = e^{\int \frac{2}{x} \, dx}$.
2. Simplify the integrating factor using log laws: $I(x) = e^{2\ln x} = e^{\ln(x^2)} = x^2$.
3. Multiply every term in the original differential equation by $x^2$: $x^2 \frac{dy}{dx} + 2xy = 4x^3$.
4. Recognize that the left-hand side is the exact derivative of the product $(y \cdot I(x))$: $\frac{d}{dx}(x^2 y) = 4x^3$.
5. Integrate both sides with respect to $x$: $x^2 y = \int 4x^3 \, dx \implies x^2 y = x^4 + C$.
6. Divide by $x^2$ to isolate $y$: $y = x^2 + \frac{C}{x^2}$.
Examiner tip: When you multiply the differential equation by the Integrating Factor, the left hand side will always compress perfectly into $\frac{d}{dx}(y \times \text{IF})$.
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.
Question 1 · easy · 4 marks · Paper 1
Verify algebraically that the function \(y = C e^{-4x}\), where \(C\) is an arbitrary constant, is a valid solution to the differential equation \(\frac{dy}{dx} + 4y = 0\).
Question 2 · medium · 6 marks · Paper 1
Find the exact particular solution to the differential equation:
\[\frac{dy}{dx} = y \cos x\]
given the boundary condition that \(y(0) = e^2\).
Question 3 · hard · 7 marks · Paper 1
A system of coupled differential equations is given by \(\vec{x}' = \begin{pmatrix} 0 & 1 \\ -2 & -3 \end{pmatrix} \vec{x}\).
The exact solution takes the form \(\vec{x}(t) = c_1 e^{\lambda_1 t} \vec{v}_1 + c_2 e^{\lambda_2 t} \vec{v}_2\).
Find the exact solution for \(\vec{x}(t)\) given the initial conditions \(x(0) = 2\) and \(y(0) = 1\).
All 19 differential equations and eulers method questions with mark schemes →
FAQ
How many IB Maths AI HL differential equations and eulers method questions are there?
There are 19 exam-style differential equations and eulers method questions in the AI HL question bank (Paper 1: 19), graded 4 easy, 4 medium, 4 hard, 7 starter. 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?
In the question bank these questions are set as Paper 1 questions.
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.
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