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' = (2 − 2ty)/(t2 + 1), 0≤ t ≤ 1, y(0) = 1, with h = 0.1; actual solution y(t) = (2t + 1)/(t2 + 1).
b. y' = y2/(1 + t), 1≤ t ≤ 2, y(1) = −(ln 2)−1, with h = 0.1; actual solution y(t) =−1/(ln(t + 1)).
c. y' = (y2 + y)/t, 1≤ t ≤ 3, y(1) = −2, with h = 0.2; actual solution y(t) = 2t/(1 − 2t).
d. y' = −ty + 4t/y, 0≤ t ≤ 1, y(0) = 1, with h = 0.1; actual solution y(t) =√((4 − 3e−t)2.)