A solution of the wave equation using D'Alembert's solution

Solve the wave equation subject to the initial conditions u(x,0)=sin(x)/(x^2+1), du/dt(x,0)=x/(x^2+1)

The Fourier coefficients of a derivative

Let f be a 2 pi periodic, differentiable function with Fourier coefficients a_n and b_n. Let (a_n)*, (b_n)* be the Fourier coefficients of f'. a) Show that (a_0)*=0 b) Use integration by parts to find a formula for the Fourier coefficients of f' in terms of the Fourier coefficients of f. (The attachment contains the above question written with clear mathematical notation)

2 examples of the method of characteristics for solving PDEs

1. Use the method of characteristics to solve the advection equation du/dt=-kdu/dx-ru subject to the initial condition u(x,0)=f(x). 2. Use the method of characteristics to solve du/dt+te^(-t^2))du/dx=usin(t) subject to the initial condition u(x,0)=e^(-x^2)) (See attachment for the above questions formatted with correct mathematical notation)

Solving the wave equation using separation of variables

See the attachment for the questions.

PDE's - separation of variables

I would like to understand how these 2 problems are solved. Thanks.