Use the methods discussed in class for the finite difference scheme and LU solver. Note that...

  

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Use the methods discussed in class for the finite difference scheme and LU solver. Note that your code should not construct the entire matrix; instead, use linear arrays containing the nonzero matrix elements. 1. Consider the 2-point BVP, -cy" + y = 2x + 1 on the domain 0 x 1, with boundary conditions y(0) = 0, y(1) = 0 and = 10-. 1- x Plot the exact solution and the numerical solution for h = 32 2. Consider the 2-point BVP Show that the exact solution is y(x) = 2x+1-( sinh +3sinh -1 v) (sinh ) . ey"+y= f(x) on x [0,1] with y(0) = 0 and y'(1) = 0. Let f(x) = (x 1) - 1. (a) Show that f(x) satisfies the boundary conditions. Find constants a, b, and c so that yp (x) = ax + bx+c satisfies the differential equation and the boundary condition at x = 1 (it doesn't have to satisfy yp (0) = 0). Activate Winc Use the methods discussed in class for the finite difference scheme and LU solver. Note that your code should not construct the entire matrix; instead, use linear arrays containing the nonzero matrix elements. 1. Consider the 2-point BVP, -cy" + y = 2x + 1 on the domain 0 x 1, with boundary conditions y(0) = 0, y(1) = 0 and = 10-. 1- x Plot the exact solution and the numerical solution for h = 32 2. Consider the 2-point BVP Show that the exact solution is y(x) = 2x+1-( sinh +3sinh -1 v) (sinh ) . ey"+y= f(x) on x [0,1] with y(0) = 0 and y'(1) = 0. Let f(x) = (x 1) - 1. (a) Show that f(x) satisfies the boundary conditions. Find constants a, b, and c so that yp (x) = ax + bx+c satisfies the differential equation and the boundary condition at x = 1 (it doesn't have to satisfy yp (0) = 0). Activate Winc