First order differential equations: Euler's method

IB Mathematics Applications and Interpretation HL· Unit 4: Calculus, Topic 12: Euler's Method· 15 min read

1. 1. Core Idea and Formula of Euler's Method★★☆☆☆HL only⏱ 5 min

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Many real-world first order differential equations do not have closed-form solutions that can be written with elementary functions. Euler's method relies on the tangent line approximation at each step to build an approximate solution curve starting from a known initial condition.

📘 Definition

Euler's Iterative Formula

$y_{n + 1} = y_{n} + h \cdot \frac{d y}{d x}_{(x_{n}, y_{n})}$

For each iteration, we use the slope of the tangent line at the current known point to estimate the value of $y$ at the next $x$ position, $h$ is the fixed step size between $x_{n}$ and $x_{n + 1}$ .

📐 Worked Example

Given $\frac{d y}{d x} = x + y$ with initial condition $y (0) = 1$ , use 2 steps of Euler's method with step size $h = 0.5$ to approximate $y (1)$ .

1
Identify initial values: $x_{0} = 0$ , $y_{0} = 1$ , $h = 0.5$ . Calculate derivative at $(x_{0}, y_{0})$ :
2
$\frac{dy}{dx}\bigg|_{(0,1)} = 0 + 1 = 1$
3
Calculate first updated values:
4
$x_1 = x_0 + h = 0 + 0.5 = 0.5 \\ y_1 = y_0 + h \cdot \frac{dy}{dx}\bigg|_{(x_0,y_0)} = 1 + (0.5)(1) = 1.5$
5
Calculate derivative at the new point $(x_{1}, y_{1})$ :
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$\frac{dy}{dx}\bigg|_{(0.5,1.5)} = 0.5 + 1.5 = 2$
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Calculate the final approximation at $x = 1$ :
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$x_2 = 0.5 + 0.5 = 1 \\ y_2 = 1.5 + (0.5)(2) = 2.5$
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Final approximation: $y (1) \approx 2.5$

2. 2. Applying Euler's Method for Multiple Steps★★★☆☆HL only⏱ 6 min

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Euler's method follows the same iterative pattern for any number of steps. IB exams almost always use equal step sizes, though the method can be adapted for variable step sizes if required. Smaller step sizes produce more accurate approximations but require more calculations.

📐 Worked Example

Given $\frac{d y}{d x} = \frac{y}{x}$ with initial condition $y (1) = 2$ , approximate $y (2)$ with step size $h = 0.25$ .

1
Initial values: $x_{0} = 1$ , $y_{0} = 2$ . Calculate initial derivative:
2
$\frac{dy}{dx}\bigg|_{(1, 2)} = \frac{2}{1} = 2$
3
Iterate for 4 total steps to reach $x = 2$ :
4
$n=1: x_1 = 1.25, \quad y_1 = 2 + (0.25)(2) = 2.5, \quad \frac{dy}{dx} = \frac{2.5}{1.25} = 2 \\ n=2: x_2 = 1.5, \quad y_2 = 2.5 + (0.25)(2) = 3, \quad \frac{dy}{dx} = \frac{3}{1.5} = 2 \\ n=3: x_3 = 1.75, \quad y_3 = 3 + (0.25)(2) = 3.5, \quad \frac{dy}{dx} = \frac{3.5}{1.75} = 2 \\ n=4: x_4 = 2, \quad y_4 = 3.5 + (0.25)(2) = 4$
5
For this DE, the exact solution is $y = 2 x$ , so the approximation here is exact.

3. 3. Error Analysis for Euler's Method★★★☆☆HL only⏱ 5 min

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All approximations from Euler's method include some error, because the tangent line at each step diverges from the true curved solution. The direction and magnitude of error follow predictable patterns that are often tested in exams.

📘 Definition

Global Truncation Error

The total error accumulated across all steps of the approximation. For Euler's method, global error is proportional to step size $h$ : halving $h$ roughly halves the total error.

📐 Worked Example

For $\frac{d y}{d x} = y^{2}$ , $y (0) = 1$ , exact solution is $y = \frac{1}{1 - x}$ . Compare error when approximating $y (0.3)$ for $h = 0.3$ vs $h = 0.1$ .

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Case 1: $h = 0.3$ (1 step):
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$y_1 = 1 + 0.3(1^2) = 1.3 \\ \text{Exact } y(0.3) = \frac{1}{1-0.3} \approx 1.4286 \\ \text{Error} = |1.4286 - 1.3| = 0.1286$
3
Case 2: $h = 0.1$ (3 steps):
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$n=0: x_0=0, y_0=1, \frac{dy}{dx}=1, y_1 = 1.1, x_1=0.1 \\ n=1: \frac{dy}{dx} = 1.1^2=1.21, y_2=1.1 + 0.1(1.21)=1.221, x_2=0.2 \\ n=2: \frac{dy}{dx} = 1.221^2 \approx 1.4908, y_3=1.221 + 0.1(1.4908) \approx 1.3701$
5
Calculate error for $h = 0.1$ :
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$\text{Error} = |1.4286 - 1.3701| \approx 0.0585$
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The error for $h = 0.1$ is roughly half the error for $h = 0.3$ , matching the proportional error relationship.

4. Common Pitfalls

Wrong move:

Rounding intermediate $y_{n}$ values to the same precision as the final answer

Why:

Rounding error accumulates across multiple steps, leading to a final result that is outside acceptable tolerance

Correct move:

Keep 1-2 extra decimal places for all intermediate values, only round the final answer

Wrong move:

Reusing the initial derivative value for all iterations

Why:

The derivative $\frac{d y}{d x}$ depends on both $x$ and $y$ , so it changes at every step

Correct move:

Recalculate $\frac{d y}{d x}$ at each new $(x_{n}, y_{n})$ before calculating $y_{n + 1}$

Wrong move:

Counting the wrong number of steps for a given interval

Why:

Many students count the starting point as a step, leading to too few or too many iterations

Correct move:

Calculate number of steps as: $n = \frac{final x - initial x}{h}$

Wrong move:

Mixing up overestimate vs underestimate for concave curves

Why:

It is easy to reverse the relationship between concavity and error direction

Correct move:

Remember: Tangent lines lie below concave up curves (underestimate) and above concave down curves (overestimate)

5. Quick Reference Cheatsheet

Concept	Rule/Formula
Iterative Formula	$y_{n + 1} = y_{n} + h \cdot f (x_{n}, y_{n}), x_{n + 1} = x_{n} + h$
Initial Condition	Start at $(x_{0}, y_{0})$ given in the question
Number of Steps	$n = \frac{x _{final} - x _{0}}{h}$
Error vs Step Size	Global error $\propto h$ → smaller $h$ = smaller error
Concave Up Solution	Euler's method gives an underestimate
Concave Down Solution	Euler's method gives an overestimate

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 2
Approximate DE solution with 3 steps
2022 · 1
Error analysis for different step sizes

Going deeper

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What's Next

Euler's method is the foundation for more advanced numerical methods for solving differential equations, which are widely used across science, engineering, economics, and data science to model dynamic systems that do not have simple analytical solutions. For IB AI HL, you will next build on this knowledge to solve first order DEs that can be solved analytically with separation of variables, then apply differential equations to model real-world dynamic systems like population growth, cooling, and chemical reactions.