Show that for any constant k, the function u(x, y) = ex cos ky is a solution of Laplace's equation Uxx + Uyy = 0. b. Show that for any constant k, the function u(x, y) = ekxek²y is a solution of the heat equation Uxx - Uy = 0. c. Show that for any constant k, the function u(x, y) = ekxe-ky is a solution of the wave equation uxx - Uyy = 0. d. Show that for any constant k, the function u(x, y) = x² + (1 - k) is a solution of Poisson's equation Uxx + Uyy = 1.