The Taylor Series Method

The Euler's method can be viewed as an approximation by the first two terms of a Taylor's series. If we calculate the higher order derivatives of y, we can write

(3.1)

We have

and so we can evaluate higher order derivatives as follows:

Writing etc., we have the next two derivatives as

(3.2)

It is immediately clear that this is not a practical method unless the function is simple enough that many of these partial derivatives vanish, but it is theoretically possible to develop as many methods like (3.1) as necessary and evaluate these derivatives for substitution in (3.1)