As with Adams formulas, modern codes based on the BDFs vary the formula (the order used) as well as the step size. The solution of problems that are quite stiff are completely impractical with a method intended for non-stiff problems, such as an explicit Runge-Kutta method or an Adams-Moulton method evaluated by simple iteration
The simplest BDF is when is the straight line interpolating and . The derivative at is the constant slope of this line and setting it to results in
Once again we have derived the backward Euler formula! Although this case results in a one-step formula, the higher order BDFs do involve previously computed solution values. For example, when the step size is a constant h, the backward differentiation formula of order two is
A code providing a highly efficient implementation of the Adam's formulas and the BDF formulas is given in the next section.