Exercise 3 (9 points) Consider the system of ODEs for the functions d(t),v(t): { mu' = -kexp(d)d cv v(0) = vo (1) . d' = vend, d(0) = do with k,c, 7 > 0. (a) 2 Show that: (v2)' + g(d)(d)' 0. (b) 1 Discretize() in time using the B-method, and denote the discrete solutions by dn = d(tn), Un = v(tn). Formulate the system of (possibly non-linear) equations for (dn+1, Un+1) as a vector root finding problem T (dn+1, Vn+1) = 0, and give the specific form for T. Assume tn+1 = to +h, h > 0. (c) 3 Write down the Newton iteration for computing the (k + 1)-st iterand #1, v*1 from the k-th iterand di +1, vh+1. Give specific expressions for the vectors and matrix involved, but you do not need to explicitly invert any matrix. (d) 3 Consider now the system of ODES: X'(t) = AX(t), X(0) = X0 = 0, (2) with A = [72 -1 -1/2 1/2 - 2 Discretize the ODE (d)) using the B-method for B = 0. Then, determine hcrit such that Xn +0 (for any Xo) when n + for h 0. (a) 2 Show that: (v2)' + g(d)(d)' 0. (b) 1 Discretize() in time using the B-method, and denote the discrete solutions by dn = d(tn), Un = v(tn). Formulate the system of (possibly non-linear) equations for (dn+1, Un+1) as a vector root finding problem T (dn+1, Vn+1) = 0, and give the specific form for T. Assume tn+1 = to +h, h > 0. (c) 3 Write down the Newton iteration for computing the (k + 1)-st iterand #1, v*1 from the k-th iterand di +1, vh+1. Give specific expressions for the vectors and matrix involved, but you do not need to explicitly invert any matrix. (d) 3 Consider now the system of ODES: X'(t) = AX(t), X(0) = X0 = 0, (2) with A = [72 -1 -1/2 1/2 - 2 Discretize the ODE (d)) using the B-method for B = 0. Then, determine hcrit such that Xn +0 (for any Xo) when n + for h