The Euler (one step method) method only uses values at
to compute
We can get better estimates for
if we make use of some of the previous values.
If
then
Using the Euler method gives
for step size
If we weight the values of
and 
then
We find
and
such that the above is exact for linear functions.
Put
respectively.
We solve the simultaneous equations
(1)
(2)
Sub (1) into (2) and
cancels to give
then from (1)
Euler's equation becomes
This is a two step method.
We can also derive a three step method.
Put
respectively.
We solve the simultaneous equations
(3)
(4)
(5)
Rearrange (4) to give
and substitute (3) to give
(6)
Rearrange (5) to give
and substitute (5) and (6) to give
(7)
(7)-(6) gives
then from (6)
and finally from (3)
Then we have
An alternative method of finding the coefficients exists using Lagrange polynomials.
is the Lagrange polynomial such that
then