Formula

Euler Rule `y_(n+1)=y_n+hf(x_n,y_n)`

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Examples
1. Find y(0.2) for `y'=(x-y)/2`, `x_0=0, y_0=1`, with step length 0.1 using Euler method (first order differential equation)

Solution:
Given `y'=(x-y)/(2), y(0)=1, h=0.1, y(0.2)=?`

Euler method

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for `n=0,x_0=0,y_0=1`

`y_1=y_0+hf(x_0,y_0)`

`=1+(0.1)f(0,1)`

`=1+(0.1)*(-0.5)`

`=1+(-0.05)`

`=0.95`

`x_1=x_0+h=0+0.1=0.1`

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for `n=1,x_1=0.1,y_1=0.95`

`y_2=y_1+hf(x_1,y_1)`

`=0.95+(0.1)f(0.1,0.95)`

`=0.95+(0.1)*(-0.425)`

`=0.95+(-0.0425)`

`=0.9075`

`x_2=x_1+h=0.1+0.1=0.2`

`:.y(0.2)=0.9075`

`n`	`x_n`	`y_n`	`x_(n+1)`	`y_(n+1)`
0	0	1	0.1	0.95
1	0.1	0.95	0.2	0.9075

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