IB Maths AA HL · Unit 5: Calculus
IB Maths AA HL Differential Equations and Maclaurin Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 49 questions
- Paper 1: 37
- Paper 2: 12
- 7 easy
- 17 medium
- 17 hard
- 8 starter
- 3 worked examples
Practise Differential Equations and Maclaurin questions →
AA HL formula booklet
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 (23)
- Maclaurin Series (20)
- Euler's Method (6)
Key formulas
- Euler's identity
- \(e^{i\pi} + 1 = 0\)
- Maclaurin series
- \(f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f'''(0)}{3!}x^3 + \cdots\)
- Standard Maclaurin expansions
- \(e^x = \sum_{k=0}^{\infty}\tfrac{x^k}{k!},\ \sin x = \sum_{k=0}^{\infty}\tfrac{(-1)^k x^{2k+1}}{(2k+1)!},\ \cos x = \sum_{k=0}^{\infty}\tfrac{(-1)^k x^{2k}}{(2k)!}\)
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 $f(x) = x\cos(2x)$.
1. Locate the standard Maclaurin series for $\cos x$ in the formula booklet: $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$
2. Substitute $2x$ in place of $x$ into the standard expansion: $\cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \dots$
3. Expand the substituted terms carefully: $\cos(2x) = 1 - \frac{4x^2}{2} + \frac{16x^4}{24} - \dots$
4. Simplify the coefficients of the cosine expansion: $\cos(2x) = 1 - 2x^2 + \frac{2}{3}x^4 - \dots$
5. Multiply the entire series by the initial $x$ multiplier: $f(x) = x\left(1 - 2x^2 + \frac{2}{3}x^4\right)$.
6. State the final expanded series up to three terms: $\mathbf{f(x) = x - 2x^3 + \frac{2}{3}x^5}$.
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 $\frac{dy}{dx} = \frac{x^2}{y}$, given the boundary condition that $y(0) = 2$. Give your answer in the explicitly solved form $y = f(x)$.
1. Separate the variables by multiplying both sides by $y$ and $dx$: $y \, dy = x^2 \, dx$.
2. Integrate both sides of the equation: $\int y \, dy = \int x^2 \, dx$.
3. Evaluate the integrals and add the constant of integration $C$ to the $x$-side: $\frac{1}{2}y^2 = \frac{1}{3}x^3 + C$.
4. Substitute the initial condition $x=0, y=2$ to solve for $C$: $\frac{1}{2}(2)^2 = \frac{1}{3}(0)^3 + C \implies 2 = C$.
5. Rewrite the particular solution implicitly: $\frac{1}{2}y^2 = \frac{1}{3}x^3 + 2$.
6. Rearrange to make $y$ the subject, taking the positive square root because the initial condition $y(0)=2$ is positive: $\mathbf{y = \sqrt{\frac{2}{3}x^3 + 4}}$.
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 $\frac{dy}{dx} + 3y = e^{2x}$. Give your answer in the form $y = f(x)$.
1. Recognize the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = 3$ and $Q(x) = e^{2x}$.
2. Calculate the integrating factor (IF) using $e^{\int P(x) dx}$: $\text{IF} = e^{\int 3 \, dx} = e^{3x}$.
3. Multiply the entire differential equation by the integrating factor: $e^{3x} \frac{dy}{dx} + 3e^{3x} y = e^{3x} e^{2x}$.
4. Condense the left-hand side using the reverse product rule: $\frac{d}{dx}(y e^{3x}) = e^{5x}$.
5. Integrate both sides with respect to $x$: $y e^{3x} = \int e^{5x} \, dx \implies y e^{3x} = \frac{1}{5} e^{5x} + C$.
6. Isolate $y$ by dividing all terms by $e^{3x}$: $\mathbf{y = \frac{1}{5} e^{2x} + C e^{-3x}}$.
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 $\frac{dy}{dx}$ 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.
Question 1 · easy · 3 marks · Paper 1
Consider the linear differential equation:
\[\frac{dy}{dx} + \frac{2}{x}y = 5x\]
Find the integrating factor required to solve this differential equation.
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 \(\frac{dT}{dt} = -k(T - T_A)\).
A cup of coffee is initially \(100^\circ\text{C}\) in a room kept at a constant \(20^\circ\text{C}\). After 10 minutes, the coffee has cooled to \(60^\circ\text{C}\).
Calculate the exact temperature of the coffee after 20 minutes.
Question 3 · hard · 6 marks · Paper 1
A function \(y = f(x)\) satisfies the differential equation:
\[(1 - x)\frac{dy}{dx} = 2y\]
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.
All 49 differential equations and maclaurin questions with mark schemes →
FAQ
How many IB Maths AA HL differential equations and maclaurin questions are there?
There are 49 exam-style differential equations and maclaurin questions in the AA HL question bank (Paper 1: 37 · Paper 2: 12), graded 7 easy, 17 medium, 17 hard, 8 starter. 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: 37 · Paper 2: 12. 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.
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