We have seen that the Euler method is not sufficiently accurate to be an efficient problem-solving procedure, so it is desirable to develop more accurate methods. Consider the initial value problem
and let 
denote its solution. By integrating the
given differential equation from 
to 
we obtain
The Euler formula
is obtained by replacing 
in eq. (11) by its approximate
value 
at the left endpoint of the interval of intergration.
