Plot the solution for k pi3 and a 1 as well as the

Question

1. Solve the initial value problem

{ u_t + cos(kt)u_x = ax²

    u(x, 0) = 0

Plot the solution for k = Π/3 and a = 1, as well as the projected characteristic curves for this PDE, finding a way to plot them both on the same graph.

2. Show that, for functions obeying homogeneous Dirichlet conditions on the boundary of a domain D, the differential operator ζ

ζ = a(x)d²/dx² + b(x) d/dx + c(x)

is self-adjoint if a' (x) = b(x).

3. Prove that if the weight function w(x) in a Sturm-Liouville problem is increased for all values of x in a domain D, then the smallest eigenvalue decreases.

4. Find all solutions to the Sturm-Liouville problem on the domain D = {lx| < d}:

[d²/dx² - β² ]u = λu on D

du/dx = 0 on ∂D

for β < Π/d. Plot the 5 solutions with the smallest values of λ. Prove that all solutions are orthogonal.