If the surface of the ball r² = x² + y² + z² ≤ R² is kept at temperature zero and the initial temperature in the ball is f(r), show that the temperature u(r, t) in the ball is a solution of u_t = c²(u_rr + 2u_r/r) satisfying the conditions u(R, t) = 0, u(r, 0) = f(r). Show that setting v = ru gives v_t = c²v_rr, v(R, t) = 0, v(r, 0) = rf(r). Include the condition v(0, t) = 0 (which holds because u must be bounded at r = 0), and solve the resulting problem by separating variables.