a backward-differentiation formula
When a problem is STIFF — meaning it has some violently fast modes mixed with slow ones — explicit methods choke, forced to take absurdly tiny steps just to stay stable. You need an implicit method tough enough to take big steps without blowing up. The backward-differentiation formulas are the implicit multistep family purpose-built for exactly this, and they are the standard engine inside stiff ODE solvers.
The idea: instead of approximating the integral of the slope (as Adams methods do), BDF fits a polynomial through the last several SOLUTION values and demands that its DERIVATIVE at the new point t_{n+1} equal the slope f(t_{n+1}, y_{n+1}). Because it uses the slope only at the new, unknown point, it is implicit — you solve for y_{n+1} each step, usually with Newton's method. The simplest, BDF1, is exactly backward Euler: y_{n+1} - y_n = h*f_{n+1}. BDF2 uses two past values: (3/2)*y_{n+1} - 2*y_n + (1/2)*y_{n-1} = h*f_{n+1}, which is second order. The family runs BDF1 through BDF6, climbing in order as it uses more history.
BDF methods earn their place by being highly stable on stiff problems: BDF1 and BDF2 are A-stable (stable for ANY decaying mode at any step size), and BDF3 through BDF6 are A(alpha)-stable (stable for everything except modes very near the imaginary axis), letting them take large steps governed by accuracy, not stability. The honest limits: by Dahlquist's second barrier, NO multistep method above order 2 can be A-stable, so BDF3-6 sacrifice a sliver of the stability region; BDF7 and beyond are not even zero-stable and are useless; and BDFs add numerical damping, so they are wrong for problems with genuine sustained oscillations. They are the right tool for stiff DECAY, not for conservative dynamics.
On the stiff system y' = -1000y + 1000 (fast relaxation to y = 1), an explicit method needs h around 0.002 to stay stable across a time span of 10 — about 5000 steps. BDF2, being A-stable, takes steps sized purely by accuracy (perhaps h = 0.1, 100 steps), each solving a small Newton system. This is why every serious stiff solver (CVODE, MATLAB ode15s) is built on BDF.
Implicit, history-using, stiff-stable — the standard engine for stiff decay.
Dahlquist's second barrier caps A-stability at order 2 for multistep methods, so only BDF1 and BDF2 are truly A-stable; BDF3-6 are merely A(alpha)-stable, and BDF orders above 6 are not zero-stable and must never be used. More history does not mean more stability.