Consider the one-dimensional wave equation Ju 1 on the interval [0, 10], subject to the boundary conditions show that = 100- u(0, t) = 0 and u(10, t) = 0, t > 0. (i) Show that a solution u(x, t) can be written as where the B, and B are constants. [50 marks] (ii) If we also have the initial conditions u(x,0) = t 2u x u(x, t) =(B, cos(nat) + B sin(nat)) sin- n=1 Min 10-z 8 (nn) 10,5x 10 0x5 (x,0)=0, 0x 10. u(x, t) = [ sin n=1 2 nax 10 -cos(nat) sin NAI 10