# What is Euler’s Method?

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

## Methodology

Euler’s method uses the simple formula,

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

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`.

In general, if you use small step size, the accuracy of approximation increases.

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

where,

• n = number of steps
• h = interval width (size of each step)

## Example

Find `y(1)`, given

Solving analytically, the solution is y = ex 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.

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.