Use Exercise 6 and the Piecewise Linear Algorithm with n = 9 to approximate the solution to the boundary-value problem
−y" + y = x, 0 ≤ x ≤ 1, y(0) = 1, y(1) = 1 + e−1.
In exercise 6
Show that the boundary-value problem
− d / dx (p(x)y') + q(x)y = f (x), 0≤ x ≤ 1, y(0) = α, y(1) = β,
can be transformed by the change of variable
z = y − βx − (1 − x)α
into the form
− d / dx (p(x)z') + q(x)z = F(x), 0≤ x ≤ 1, z(0) = 0, z(1) = 0.