**Overview**

This method was given by Leonhard Euler. Euler’s method is the first order numerical methods for solving ordinary differential equations with given initial value.

It is the basic explicit method for numerical integration of the ODE’s.

**Euler method**

The general first order differential equation

With the initial condition

In this method the solution is in the form of tabulated values.

Integrating both sides of the equation (i) we get

Assuming that in ,this gives Euler’s formula

In general formula

Error estimate for the method

**Example: Use Euler’s procedure to find y(0.4) from the differential equation**

Sol:

Given equation dy/dx = xy

Here

We break the interval in four steps.

So that

By Euler’s formula

For n=0 in equation (i) we get, the first approximation

n=1 in equation (i) we obtain

Put=2 in equation (i) we get, the third approximation

Put n=3 in equation (i) we get, the fourth approximation

Hence y(0.4) =1.061106.

**Modified Euler’s Method:**

Instead of approximating f(x, y) by as in Euler’s method. In the modified Euler’s method we have the iteration formula

Where is the nth approximation to y1 .The iteration started with the Euler’s formula

**Example: Use modified Euler’s method to compute y for x=0.05. Given that**

**Result correct to three decimal places.**

**Sol:**

Given equation dy/dx = x + y

Here f(x, y) = x + y with y(0) = 1

Take h = 0.05 – 0 = 0.05

By modified Euler’s formula the initial iteration is

The iteration formula by modified Euler’s method is

For n=0 in equation (i) we get

Where as above

For n=1 in equation (i) we get

For n=3 in equation (i) we get

Since third and fourth approximation are equal .

Hence y=1.0526 at x = 0.05 correct to three decimal places.

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