Here is a problem in Partial Diff Q related to the wave equation

2. Let n{z.t} = so: +e} - so a}. so} = % fa g\"[;i:)di, where g\" is the odd Eli-periodic extension of 9. Show that n is the solution of the vibrating string problem with boundary conditions n{,i} = [II and nIIL't} = I] and initial conditions shall} = Ill and ndsa] = 9(3)} [a] Let 90(3) he odd and Ell-periodic. Use these properties to derive the identity ll - s} = -s"lL + s}- This identity will he applied to the function 9" to show that nlL. t) = l]. (b) Show that 1: satises the initial conditions 14(1', ll] = l} and u; (3:, {1] = 9(3) on the interval neeeh [c] Show that 1.: satises the differential equation on = sin\". [d] Show that it satises the boundary condition EU}, t] = ll. [e] Show that n satises the boundary condition n(L,t) = If]. This is tricky. The easiest way to do it is to start by combining C(L + ct) G(L er] into a single integral and then partioning that integral into a portion from L oi". to L and a portion from L to L + ct. Then do a change of variables in each of the integrals, using 1; = L i in the rst integral and y = i: L in the second one. Apply the identity from {a} to one of these integrals, and the result follows immediately