length, is given by the heat equation, ди 2²u əx² Ət (0 < x < 1, t > 0), here x measures distance along the bar, t is time, and u is temperature easured in ° C. Initially the bar has the temperature distribution (0 < x < 1). 2 u(x,0) = cos² 5πα 2 sin ( 5πα 2 he end of the bar at x = 0 is kept in contact with a block of ice at C, while the other end at x 1 is insulated. = O Write down the boundary conditions for the temperature distribution that model the situation described above. ) Use trigonometric identities to show that the initial condition can be expressed as u(x, 0) = P sin 5пх 7²) + 2 + Q sin (²¹ 15πα 2 and find the real numbers P and Q. Consider the eigenvalue problem X"(x) + XX(x) = 0, X(0)= X'(1) = 0. Find the eigenvalues and eigenfunctions. You may assume that all eigenvalues are positive. ) Use the method of separation of variables to find an expression for the temperature distribution in the bar for t > 0.