Consider the PDE given by u"(x) = sin(x) on [0, 1] such that u(0) = 0 and u(1) = 1. (a) (5 points) Solve this boundary value problem analytically. (b) (10 points) Using the second-order approximation of the second derivative, write the associated linear algebra problem that can be solved to approximate the solution to this PDE for a given value of h. For [a, b], use gridpoints located at xo = a, x1 = a +h, x = a + 2h, ...N = b - h, xN+1 = b. Therefore, your unknowns will be u,..., un, and you b will have h N + 1 (c) (20 points) Solve this PDE in Matlab with Matlab's backslash for values of N = 23, 24,..., 22. Perform a refinement study, using the relative error of your solution, in the max norm. In your refinement study log log plot, use N for the x-axis. (This is more natural than the use of h that we must use for problem (3). This is because convergence shows error going down as you increase N, left to right.) What order of convergence do you get? Also plot your approximate solutions and the true solution.