Backward  Method :

Instead of using Taylor's formula at the point x₀+h one can also use x₀-h and develop Euler's backward formula in the following manner. Consider the Taylor series at x = x₀ - h

 y(x₀ - h) = y(x₀) - hy'(x₀) + (h²/2) y''(x₀) + . . .

If we truncate the infinite series on the right hand side after second term, we can write
 

	y(x0 - h) = y(x0) - hy'(x0)
	-y(x0) = -y(x0 - h) - h f(x0, y0)
or	yi = yi-1 + h fi	for any i

since f_i = f(x_i, y_i) has also the unknown y_i, the formula has more than one term which has the unknown and hence the scheme is an implicit one.
Local truncation error is (h²/2) y''(x) where x_i < x < x_i+1.

That is, the order of the scheme is same as Euler forward scheme, which is very convenient to use because of its explicit nature whereas the backward scheme is an implicit one in which an approximation to y_i has to be computed with help of some other scheme. Hence using Euler Forward scheme is time saving process than Euler backward scheme with the same order of accuracy.
 

Mid-point Method :

Euler's forward and backward schemes 

y_i+1 = y_i + h f(x_i, y_i)
y_i = y_i-1 + h f(x_i, y_i)

can be clubbed together to get a second order scheme of the form

y_i+1 = y_i-1 + 2h f(x_i, y_i)

The only drawback of this scheme is at i = 0, the scheme uses the value of the dependent variable at i = -1 which is not available. Hence one has to use some other scheme to compute the value of the dependent variable at first node (next to the initial point). However from second node onwards the solution can be obtained with second order accuracy using the mid point Nystrom method.
 

Local truncation error : 

T_i+1 = y(x_i+1) - y_i+1
        = y(x_i+1) - y_i-1 + 2h f(x_i, y_i)
       = h³/3 y'''(x)         where x_i < x < x_i+1. 

or |T_i+1| <  h³/3 M 
 

where M =	max	|y'''(x)|
	x Î (x0, xn)