Quasi Linear PDE

See Attached problem.

Initial Value Problem + Method of Characteristics

Solve the following initial data problem: u_x + u_y = u^2 u(x,0) = h(x) I have that x_t = 1, y_t = 1 and z_t = z^2 also, x(0) = s, y(0) = 0 and z(0)=h(s) from this I have x=s + t and y=t please provide a detailed solution of how to find z.

PDE Method of Characteristics

I have attached a problem along with a solution I found (I do not know if this solution is accurate). Can you please explain the solution to me, including details?

Tempered Distribution

Show that exp(x) is not a tempered distribution. Please justify your steps. Thank you

Fourier Transform

Compute the 1-dimensional FT of 1/(1+x^2)^k by applying the calculus of residues.