math expert required

Problem: (20 points) Consider the following two-point boundary value problem (BVP) y"(x) -xy(x) - (1+x)y(x) =0, y(0) =1, y(1) =2. (1) (1) Show that the problem (1) admits a unique solution. (2) Describe the shooting method for solving the two-point BVP (1) Divide the interval [0, 1] uniformly into (n + 1) sub-intervals each of length / and let xi = ih, i =0, 1,...,n + 1. Denote by y, the approximation of y(x). (3) Using the shooting method with n = 1 and invoking explicit Euler method for the associated initial value problems with y'(0) = 0 and y'(0) = 1, respectively, compute the approximate value of y(1/2). (4) Describe the finite difference method for solving the BVP (1). (5) Write explicitly the tridiagonal linear system Ay = b, where y = (y1,)2,...,>n). obtained from the finite difference method. (6) Give a sufficient condition on the step size h for which the above linear system admits a unique solution. (7) Show that the global error En = maxi-0,1,..my(x,) -ya| of the finite difference method is in O(12). (8) Compute the approximate value of y(1/2) obtained from the finite difference method with n = 2. Scanned with Ca