7 (Heat equation) Consider the heat equation K- 82b მს მ2 Ət (8) where is a positive constant, E [0, a] and is real-valued. For t> 0, we demand Dirichlet boundary conditions (t, 0) = 0 at x=0 (where 0 is a constant) and von Neumann boundary conditions (t,a) = 0 at x = a. The initial condition is (0,x) = 0. (a) As a preparation, we introduce the functions gk : [0, a] → R defined by Ik (x) = √ sin(962), = gk (9) ==== where k 0,1,... Show that these functions satisfy the boundary conditions gk (0) = 0 and g(a) 0, that they form an ortho-normal system relative to the standard scalar product on L²([0, a]) and that they are eigenfunctions of with d² dx-29k = -99k (10)