Another type of multistep method arises by using a polynomial 
to
approximate the solution 
of the initial value problem rather than
its derivative 
, as in the Adams methods. We them
differentiate 
and set 
equal to 
to obtain an implicit formula for 
. These are called
backward differentiation formulas.
The simplest case uses a first degree polynomial 
. The
coefficients are chosen to match the computed values of the solution
and 
. Thus A and B must satisfy
Since 
, the requirement that
is just
Another expression for A comes from subtracting the first of Eqs. (5) from the second, which gives
Substituting this value of A into Eq. (6) and rearranging terms, we obtain the first order backward differentiation formula
Note that this equation is the backward Euler formula, we had seen earlier.
By using higher order polynomials and correspondingly more data point one can obtain backward differentiation formulas of any order. The second order formula is
and the fourth order formula is
These formulas have local truncation error proportional to 
and 
,
respectively.
