---
title: Euler's Method
---

# Euler's Method

The Euler's method is a first-order numerical procedure for solving ordinary differential equations (ODE) with a given initial value.

## The General Initial Value Problem

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn006.png)

## Methodology

Euler's method uses the simple formula,

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn3.png)

to construct the tangent at the point `x` and obtain the value of `y(x+h)`, whose slope is, 

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn008.png)


<img src="https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/Euler.png" width="600">


In Euler's method, you can approximate the curve of the solution by the tangent in each interval (that is, by a sequence of short line segments), at steps of `h`.

<i>In general</i>, if you use small step size, the accuracy of approximation increases.

## General Formula

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn7.png)

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn_new_2.png)

## Functional value at any point `b`, given by `y(b)`

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn6.png)

where,
* <b>n</b> = number of steps
* <b>h</b> = interval width (size of each step)

### Pseudocode

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn_new_1.png)

## Example

Find `y(1)`, given

&nbsp;&nbsp;&nbsp;&nbsp; ![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/eqn007.png)


Solving analytically, the solution is <i><b>y = e<sup>x</sup></b></i> and `y(1)`= `2.71828`. (Note: This analytic solution is just for comparing the accuracy.)

Using Euler's method, considering `h` = `0.2`, `0.1`, `0.01`, you can see the results in the diagram below.

![](https://raw.githubusercontent.com/pranabendra/articles/master/Euler-method/images/comparison.png)

When `h` = `0.2`, `y(1)` = `2.48832`  (error = 8.46 %)

When `h` = `0.1`, `y(1)` = `2.59374`  (error = 4.58 %)

When `h` = `0.01`, `y(1)` = `2.70481`  (error = 0.50 %)

You can notice, how accuracy improves when steps are small.

## More Information:
1. <a href='http://calculuslab.deltacollege.edu/ODE/7-C-1/7-C-1-h-c.html' target='_blank' rel='nofollow'>Numerical Methods for Solving Differential Equations</a>
2. <a href='https://en.wikipedia.org/wiki/Euler_method' target='_blank' rel='nofollow'>Euler's method</a>