A BDF formula is obtained by first writing the ODE on a point t_n+k as y' (t_n+k) = f(t_n+k, y_n+k). Then the unknown function y is interpolated at the left hand side of y' = f(t, y) using the nodes t_n,........,t_n+k and then differentiated. Obtain the BDF2 formula (k = 2)and find its order of accuracy.

Prove that BDF2 is absolutely stable in the real interval (-, ). (Hint: the characteristic polynomial for this method applied to the test equation is

t²(3-2 ) -4t +1 = 0

with roots

t_{12 = (2}1+2)/3-2

if -1/2<= < 0 then t₁₂ are both real and ... If, however, < 1/2 then t₁₂ is complex and

modulus is ...)