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 ? ? (-?, 0). (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 -1/212 are both real and ... If, however, ? 12 is complex and

modulus is ...)

10. A BDF formula is obtained by first writing the ODE on a point trek as y'(trek) = f(tn+k, Un+k). Then the unknown function y is interpolated at the left hand side of y' = f(t, y) using the nodes tn, ..., tn+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 h e (-co, 0). (Hint: the characteristic polynomial for this method applied to the test equation is 12 (3 - 2h) - 4t + 1 = 0 with roots t12 = 2+ V1+ 2h 3 - 2h if -