Use the Modified Euler method to approximate the solutions to each of the following initial-value problems, and compare the results to the actual values.
a. y' = et−y, 0≤ t ≤ 1, y(0) = 1, with h = 0.5; actual solution y(t) = ln(et + e − 1).
b. y' = (1 + t)/(1 + y), 1≤ t ≤ 2, y(1) = 2, with h = 0.5; actual solution y(t) =√(t2 + 2t + 6−1).
c. y' = −y + ty1/2, 2 ≤ t ≤ 3, y(2) = 2, with h = 0.25; actual solution y(t) =(t − 2 +√2ee−t/2)2.
d. y' = t−2(sin 2t − 2ty), 1 ≤ t ≤ 2, y(1) = 2, with h = 0.25; actual solution y(t) =1/2 t−2(4 + cos 2 − cos 2t).