Euler's Method (AI HL)
Euler's method builds an approximate numerical solution to a first-order differential equation one small step at a time, using only the gradient at your current point. It's the workhorse behind slope-field pictures and the coupled-system models later in the course. This page covers the update-step formula, how step size affects accuracy, and the mistakes that cost marks. It's part of the broader Differential Equations topic.
11 questions on this sub-topic.
The update-step formula
Covered under IB syllabus reference AHL5.16: Euler's method for finding the approximate solution to first-order differential equations of the form \(\dfrac{dy}{dx}=f(x,y)\). Spreadsheets or lists should be used to generate approximate solutions; in exams, values are generated using permitted technology.
Euler update step
\(y_{n+1}=y_n+h\,f(x_n,y_n)\)
This is in the formula booklet. \(h\) is the step size and \(f(x_n,y_n)\) is the gradient evaluated at the point you've just reached - not the point you're heading to.
Comparing to an exact solution
\(\%\text{ error}=\dfrac{|\text{exact}-\text{estimate}|}{\text{exact}}\times100\)
The general percentage-error formula, not specific to this topic's booklet section, but useful when a question asks how good an Euler estimate is.
Need the full syllabus wording and the wider calculus formula table? See Differential Equations.
Worked examples
Euler's method is used to approximate the solution of \(\dfrac{dy}{dx} = x + y\) with step size \(h = 0.1\), starting at \((0, 1)\).
Find the approximate value of \(y\) after one step.
Worked solution
\(y_1 = y_0 + h\,f(x_0, y_0).\) M1
\(f(0,1) = 1.\) A1
\(y_1 = 1 + 0.1(1)\) M1 \(= 1.1.\) A1
Use Euler's method with step \(h=0.5\) to estimate \(y(1)\) given \(\tfrac{dy}{dx} = x + y\), \(y(0)=1\).
Worked solution
The iteration is \(y_{n+1} = y_n + h\,f(x_n, y_n)\), with \(f(x,y) = x + y\). M1
First step, from \(x_0=0, y_0=1\): \(y_1 = 1 + 0.5(1) = 1.5\) at \(x_1 = 0.5\). A1
Second step: \(y_2 = 1.5 + 0.5(0.5 + 1.5) = 2.5\) at \(x_2 = 1\). A1
Common mistakes
- Using the updated \(y\)-value too early. Each step evaluates \(f(x_n,y_n)\) using the value from the previous step, not a value you've already updated within the same step.
- Losing track of \(x_n\) as well as \(y_n\). \(x\) increases by \(h\) at every step too - forgetting to update it means you evaluate the gradient at the wrong point on the second and later iterations.
- Rounding intermediate values too early. Carry full precision through each step and only round the final answer - rounding \(y_1\) before computing \(y_2\) compounds the error over multiple steps.
Ready to practise properly?
11 Euler's-method questions, marked instantly like the real exam.
Quick answers
What is the formula for Euler's method?
\(y_{n+1} = y_n + h\,f(x_n, y_n)\), where \(h\) is the step size and \(f(x_n, y_n)\) is the gradient at the current point.
Does a smaller step size always give a better estimate?
Generally yes - a smaller \(h\) reduces the truncation error at each step, giving a more accurate approximation, but it also means many more steps and more accumulated rounding error.
Where do I do this on my GDC?
See the Using your GDC section on the Differential Equations page for calculator-specific steps.