---
title: "Euler's method explain"
subject: "Numerical Method"
chapter: 6
key: "semester_4/numerical_method:euler-formula-derive"
source: https://www.bctengineer.com/q/semester_4/numerical_method/euler-formula-derive
site: BCT Engineering
---

# Euler's method explain

Numerical Method · Chapter 6: Solution of ordinary differential equations

- Asked in: 2072 Asoj, 4 marks; 2070 Magh, 4 marks (asked 2 times)
- Video: [Euler method | Lecture 48 | Numerical Methods for Engineers](https://www.youtube.com/watch?v=WMQ2Cac4sqw) — Jeffrey Chasnov
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/euler-formula-derive
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-6

## Question

Derive Euler's formula for solving an initial value problem (explain the technique of solving an initial value problem using Euler's method).

## Short answer

Euler's method is derived from Taylor series by neglecting O(h²) terms: y_{n+1}=y_n+h f(x_n,y_n). It has local error O(h²) and global O(h).

## Answer

Euler's method is the simplest one-step method for the initial value problem dy/dx = f(x, y), y(x₀) = y₀. It advances the solution along the tangent line at each point.

#### Derivation (Taylor series)
Divide the range into steps of width h, so xₙ₊₁ = xₙ + h. Expand y(xₙ + h) by Taylor's series about xₙ:

```text
y(xₙ + h) = y(xₙ) + h·y'(xₙ) + (h²/2!)·y''(xₙ) + …
```

Neglect terms of order h² and higher. Since y'(xₙ) = f(xₙ, yₙ):

```text
yₙ₊₁ = yₙ + h·f(xₙ, yₙ)        ← Euler's formula
```

Geometrically: the curve between xₙ and xₙ₊₁ is replaced by its tangent at (xₙ, yₙ), whose slope is f(xₙ, yₙ).

```text
   y
   |         .  true curve
   |      .  _/ yₙ₊₁ (Euler)
   |   . _/
   | yₙ/
   +-----|--------|------ x
        xₙ      xₙ₊₁
```

#### Procedure
1. Given f(x, y), x₀, y₀, step h and the target x.
2. Compute the slope f(xₙ, yₙ) at the present point.
3. Compute yₙ₊₁ = yₙ + h·f(xₙ, yₙ) and set xₙ₊₁ = xₙ + h.
4. Repeat steps 2–3 until x is reached.

#### Example
dy/dx = x + y, y(0) = 0, h = 0.1: y₁ = 0 + 0.1(0 + 0) = 0.0000; y₂ = 0 + 0.1(0.1 + 0) = 0.0100.

#### Error
The local truncation error is O(h²) and the global error is O(h), so the method is first-order accurate. Small h is needed for good accuracy; the method may be unstable for stiff equations. Using the average of two slopes gives the improved (Heun) and Runge–Kutta methods.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is Euler's formula derived?
   A. Integrate the Simpson polynomial
   B. Average two slopes
   C. Truncate Taylor series after the first-order term
   D. Solve a linear system
   Answer: C. y(xn+h) ≈ y(xn) + h y'(xn). (Key point: Derivation)

2. Which is Euler's formula?
   A. y(n+1) = y(n) + h f(x(n+1), y(n+1))
   B. y(n+1) = y(n) + h f(xn,yn)/2
   C. y(n+1) = y(n) + f(xn,yn)/h
   D. y(n+1) = y(n) + h f(xn, yn)
   Answer: D. Slope at the present point times h. (Key point: Formula)

3. What does one Euler step do geometrically?
   A. Follows the tangent at (xn, yn)
   B. Follows a chord
   C. Follows a parabola
   D. Jumps to exact curve
   Answer: A. The curve is replaced by its tangent. (Key point: Geometry)

4. What is the order of Euler's method?
   A. Second order
   B. First order: local O(h²), global O(h)
   C. Fourth order
   D. Zero order
   Answer: B. Neglected terms are O(h²) per step. (Key point: Error)

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