Although q(x) < 0 in the following boundary-value problems, unique solutions exist and are given. Use the Linear Shooting Algorithm to approximate the solutions to the following problems, and compare the results to the actual solutions.
a. y" + y = 0, 0 ≤ x ≤ π/4, y(0) = 1, y(π/4) = 1; use h = π / 20; actual solution y(x) = cos x + (√2 − 1) sin x.
b. y" + 4y = cos x, 0 ≤ x ≤ π/4, y(0) = 0, y(π/4) = 0; use h = π/20; actual solution y(x) = −1/3 cos 2x − √2/6 sin 2x + 1/3 cos x.
c. y" = −4x−1y' − 2x−2y + 2x−2 ln x, 1≤ x ≤ 2, y(1) = 1/2, y(2) = ln 2; use h = 0.05; actual solution y(x) = 4x−1 − 2x−2 + ln x − 3/2.
d. y" = 2y' − y + xex − x, 0≤ x ≤ 2, y(0) = 0, y(2) = −4; use h = 0.2; actual solution y(x) = 1/6 x3ex - 5/3 xex + 2ex − x − 2.