PROBLEM The oscillations of an elastic string with damping are modelled by: 1 d’u ди ди +2y: = 0 in (0,L)×(0,+∞), c? ôt? ôt ôx? (1) u(0,t) = u(L,t)= 0,t > 0, %3D ди u(x,0) = u,(x),(x,0) = u, (x),0< x < L, ốt where in (1): (a) c and y are two positive constants. (b) uo and u1 are two given functions of x. 1) Show that if system (1) has a solution it is necessarily unique. 2) Assuming that L = 1, c = 1, y = 1, and uo(x) = sinx, u1(x) = 1, V x e (0, 1), use the method of separation of variables to solve the related system (1). 3) Same question than 2), excepted that the condition y = 1 is replaced by y = x.

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Pleae refer to part (a) to answer this question.

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