Solve the following stiff initial-value problems using Euler's method, and compare the results with the actual solution.
a. y' = −9y, 0≤ t ≤ 1, y(0) = e, with h = 0.1; actual solution y(t) = e1−9t .
b. y' = −20(y−t2)+2t, 0≤ t ≤ 1, y(0) = 1/3 , with h = 0.1; actual solution y(t) = t2+1/3 e−20t .
c. y' = −20y + 20 sin t + cos t, 0 ≤ t ≤ 2, y(0) = 1, with h = 0.25; actual solution y(t) = sin t + e−20t
d. y' = 50/y−50y, 0≤ t ≤ 1, y(0) = √2, with h = 0.1; actual solution y(t) = (1+e−100t)1/2.