Use the Piecewise Linear Algorithm to approximate the solutions to the following boundary-value problems, and compare the results to the actual solution:
a. −x2y" − 2xy' + 2y = −4x2, 0 ≤ x ≤ 1, y(0) = y(1) = 0; use h = 0.1; actual solution y(x) = x2 − x.
b. - d/dx (exy') + exy = x + (2 − x)ex, 0 ≤ x ≤ 1, y(0) = y(1) = 0; use h = 0.1; actual solution y(x) = (x − 1)(e−x − 1).
c. - d/dx (e−xy') + e−xy = (x − 1) − (x + 1)e−(x−1), 0 ≤ x ≤ 1, y(0) = y(1) = 0; use h = 0.05; actual solution y(x) = x(ex − e).
d. −(x + 1)y" - y' + (x + 2)y = [2 − (x + 1)2]e ln 2 − 2ex, 0 ≤ x ≤ 1, y(0) = y(1) = 0; use h = 0.05; actual solution y(x) = ex ln(x + 1) − (e ln 2)x.