2. Given an ODE y'(t) = f(y(t), t), a 0-scheme for fixed 0 e [0, 1], is the discretization scheme (un is an approximation of y(nAt)): Un+1 = Un + At ((1 - 0) f(un, tn) + 0 f(un+1, tn+1)). (1) (a) For which values of 0 do we recover Forward Euler ? Backward Euler ? Crank-Nicolson ? (b) Consider the initial value problem y' = ly, y(0) = 1, satisfying limt-. y(t) = 0, and given 0 E [0, 1] and At > 0, let un the sequence generated by the 0-scheme generally defined above. Give a general expression of un in terms of n, 1, 0 and At. (c) For which values o do we have unconditional stability ? (i.e., limn + Un = 0 regardless of At). (d) For the other values of 0, give the interval of stability (i.e., the interval of At for which limn-+0 Un = 0)