Problem 1 (d'Alembert's solution, Haberman 4.4.6-8) Consider the initial value problem for the 1-D wave equation on R x (0, 00) u( x, 0) = uo(x), TER 1 (r, 0) = 1(T), TER where up, vo E C'(R) are two given functions. (a) Let Lu = u, tou, and Mu = u -out. Use the method of characteristics (twice) applied to the factorization Ou = MLu = 0 to obtain d'Alembert's solution u(x, t) = =[uo(x - of) + uo(x + ob)] + 5/ 20(5) dE. - ot (b) Show that if uo and vo are both odd and periodic with period 2L, then the re- striction of d'Alembert's solution to [0, L] x (0, co) satisfies the initial/boundary value problem on R x (0, 00) 1(0, t) = 0 = u(L,t), t20 u(x, 0) = uo(), TER t (x, 0) = vo(x). TER (c) Consider the initial/boundary value problem on R x (0, Do) u(0, t) = 0 = u(L,t), t>0 u(x, 0) = 9(x), TER w(x, 0) = h(x), TER where g, h E C'[0, LJ. Can d'Alembert's solution be applied to solve this prob- lem? Why or why not