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

eqn006

Methodology

Euler’s method uses the simple formula,

eqn3

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

eqn008
Euler

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

eqn7
eqn_new_2

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

eqn6

where,

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

Pseudocode

eqn_new_1

Example

Find y(1), given

eqn007

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.

comparison

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.