Although q(x) < 0 in the following boundary-value problems, unique solutions exist and are given. Use the Linear Finite-Difference Algorithm to approximate the solutions, 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, 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.
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