Second-order backward differentiation formula ($3y_{n+2}-4y_{n+1}+y_n=2hf(y_{n+2})$)

The displayed implicit linear multistep method is second order and A-stable. It is the $\alpha =4/3$ member of the trapezoidal-BDF two-step family, whose order and boundary-locus proof establish both properties. Its derivative formula uses three equally spaced values; changing the steps requires different coefficients.