Use the Nonlinear Finite-Difference Algorithm with TOL = 10−4 to approximate the solution to the following boundary-value problems. The actual solution is given for comparison to your results.
a. y" = −e−2y, 1≤ x ≤ 2, y(1) = 0, y(2) = ln 2; use N = 9; actual solution y(x) = ln x.
b. y" = y' cos x−y ln y, 0 ≤ x ≤ π/2, y(0) = 1, y (π/2) = e; use N = 9; actual solution y(x) = esin x.
c. y" = − (2(y')3 + y2y') sec x, π/4 ≤ x ≤ π/3, y (π/4) = 2−1/4, y (π/3) = 1/2 4√12; use N = 4; actual solution y(x) = √sin x.
d. y" = 1/2 (1 − (y')2 − y sin x), 0≤ x ≤ π, y(0) = 2, y(π) = 2; use N = 19; actual solution y(x) = 2 + sin x.