The Euler (one step method) method only uses values atto computeWe can get better estimates forif we make use of some of the previous values.

Ifthen

Using the Euler method givesfor step sizeIf we weight the values ofand then

We findandsuch that the above is exact for linear functions.

Putrespectively.

We solve the simultaneous equations

(1)

(2)

Sub (1) into (2) andcancels to givethen from (1)

Euler's equation becomesThis is a two step method.

We can also derive a three step method.

Putrespectively.

We solve the simultaneous equations

(3)

(4)

(5)

Rearrange (4) to giveand substitute (3) to give(6)

Rearrange (5) to giveand substitute (5) and (6) to give(7)

(7)-(6) givesthen 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 thatthen