For a small step size , the derivative is close enough to the ratio . In the Euler's method, such an approximation is attempted. To recall, we consider the problem (). Let be the step size and let with . Let be the approximate value of at . We define

The method of determination of by () is called the EULER'S METHOD. For convenience, the value is denoted by , for .

- Euler's method is an one-step method.
- The Euler's method has a few motivations.
- The
derivative
at
can be approximated by
if
is sufficiently small. Using such an approximation in (), we have
- We can also look at
() from the following point of view. The integration of () yields
- Moreover, if
is differentiable sufficient number of times, we can also arrive at
() by considering the Taylor's expansion

- The
derivative
at
can be approximated by
if
is sufficiently small. Using such an approximation in (), we have

We illustrate the Euler's method with an example. The example is only for illustration. In (), we do not need numerical computation at each step as we know the exact value of the solution. The purpose of the example is to have a feeling for the behaviour of the error and its estimate. It will be more transparent to look at the percentage of error. It may throw more light on the propagation of error.

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We now give a sample flow chart for the Euler's method.

A K Lal 2007-09-12