Please solve each and include short explanation Thank you T

T F The wave equation (u_tt = KU_XX, u(0,t) = u(L, t) = 0, u(x,0) = f(x), U_t(x, 0) = g(x), always has a steady state solution. (b) T F if f(x) is an odd periodic function defined on the interval [- L, L] then the coefficients of the sine terms in its Fourier Series are all equal to zero. T F If S(x) is the Fourier Series of f(x) = {x+1:-1 lessthanorequalto x lessthanorequalto 1/2 x-1:1/2

Solution

(a) True. The given equation represents one dimensional wave equation. No displacement at the end points x=0, x=L.

the intial displacement f(x) and initial veloicit g(x).

(b) False. The cosine terms are zero but not sine terms

(c) True

(d) True

(e) true. convolution theroem.