Solving wave equation

Question

Assignment:

Q1. Solve the wave equation,

∂2u/∂t2 = c2(∂2u/∂x)    -∞ < x < ∞

With initial conditions, u(x,0) = (1/x²+1)sin(x), and ∂u/∂t(x,0) = x/(x²+1)

Q2. Suppose that f is a 2?-periodic differentiable function with Fouier coefficients a0, an and bn.  Consider the Fourier coefficients of f ' given by

a₀ = 1/2?∫^?_-? f '(x) dx,   a_n = 1/? ∫^?_-? f '(x) cos(nx) dx,     b_n = 1/? ∫^?_-? f '(x) sin(nx) dx,    

a) Show that a₀ = 0.

b) Using integration by parts on the formula for a_n and b_n, find a formula for the Fourier coefficients of f ' in terms of the Fourier coefficients of f.

Provide complete and step by step solution for the question and show calculations and use formulas.