Methods based on numerical integration:

An exact solution of the differential equation by definition satisfies the identity

for any two points and in the interval . The methods now to be discussed are based on replacing the function , which is unknown, by an interpolating polynomial having the values on a set of points where has already been computed or is just about to be computed, evaluating the integral and accepting its value as the increment of the approximate values between and . We shall assume that the interpolating points are . To approximate by an interpolating polynomial through the above values, we use the Newton backward difference formula. If has a continuous the derivative, and backward differences are given by

where , then