In this problem, y = c1ex + c2e-x is a two-parameter family of solutions of the second-order de y'' - y = 0. find a solution of the second-order ivp consisting of this differential equation and the given initial conditions. y (0) = 1, y' (0) = 8

y (1) = 0 y' (1) = e y" = c1ex + c2e-x y' = c1ex - c2e-x for solving c1, 0 = c1e1 + c2e-1 this implies that c1 = - (c2/e2) and to solve c2 e1 = (-c2e-2) e1 - c2e-1 e1 = (-2c2e-1) c2 = - (e1/2e-1) = - (e2/2) c1 = - (c2/e2) = (e2/2e2) Therefore y = (e2/2e2) ex - (e2/2) e-x