5. * Consider the following system ut - (u" ) xx = 0, u20, xER, t>0 with the following conditions ux (t, 0) = 0, lim u = 0, u(t, x) dx = 1 202-+00 - oo (a) What is the Order of the above equation? Is it linear or non-linear? Is it homogeneous or non- homogeneous? (b) Guess a solution of the form u(t, x) = t-of (n) where n = act-P. Use chain rule to determine the values of a and S which reduce the equation to an (2 fin + 3 1 ) = 0 (c) Solve this equation and determine u(t, x). Hint: u(t, x) is a piecewise continuous function. (d) Sketch the solution when t =1,2,3