Consider a thin insulated metal rod of length 1, which satisfies the differential equation a
∂θ/∂t = ∂^2 θ / ∂x^20 < x < 1, t > 0. at Initially at t = 0, the temperature of the rod is given by θ (x,0) = f (x). Then the left end is placed in an ice bath and held at 0°C, and the right end is insulated. Use separation of variables (x, t) = v(x) w (t) to reduce this PDE to the system v (2) d2 v (2) dc2 d w (t) = dt dw (t). Find all eigenvalues k and eigenfunctions uk (2) that satisfy the boundary conditions specified in this problem for k=0,1,2,.... For k = 0,1,2,3,..., dk For k = 0,1,2,3,..., Uk (2)