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) 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 h and let x; = ih, i = 0,1,...,n+1. Denote by yn the approximation of y(xn). (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 = (91,92,... yn), 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... y(xn) - Yn of the finite difference method is in 0(h). (8) Compute the approximate value of y(1/2) obtained from the finite difference method with n=2. 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) 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 h and let x; = ih, i = 0,1,...,n+1. Denote by yn the approximation of y(xn). (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 = (91,92,... yn), 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... y(xn) - Yn of the finite difference method is in 0(h). (8) Compute the approximate value of y(1/2) obtained from the finite difference method with n=2