**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.

**The General Initial Value Problem**

**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.

**General Formula**

**Functional value at any point **`b`

, given by `y(b)`

`b`

, given by `y(b)`

where,

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

**Pseudocode**

**Example**

Find `y(1)`

, given

Solving analytically, the solution is *y = e^{x}* 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.