1. Consider the following second order partial differential equation for u, (1 + x)uxx - 2xylazy - 17 ugu = 0, for (x, y) E R2. Find the regions in the xy-plane where the above equation is elliptic, hyperbolic, or parabolic. 2.* Let u(x) = log(x,+x) for x = (X1, X2) E R2\\((0, 0)). Define B(0;2) =: ((x1, X2) E R2 : x7+x? 0 u(x, 0) = 1, lim u(x, y) =0, 0 0. 4.* Let f E L'(R) and h(x) = f(x + a) for constant a E R and for any x E R. Find the Fourier transform of h in terms of f