The faces x = 0 and x = c of a slab 0 < x < c, which is initially at temperatures f(x), are kept at temperature zero. Use the following method to derive an expression for the temperatures u = u(x, t) throughout the slab when i > 0. (a) After writing the boundary value problem for the temperatures, make the substitution s = Tx/c to show that u, = (km?/c?)u,, u = 0 when s 0 and s = T, and u - f(cs/T) when t= 0. (b) By referring to the solution (5), with coefficients (7), of the problem in Example 1, Sec. 32, write an expression for u in terms of s and t. Then, with the aid of the relation s = Tx/c that was used in part (a), show that 4%3D u(x, 1) = Eb, exp - "** n??k t sin n-1 where sin dx (n = 1, 2,...).