Show that each of the following initial-value problems has a unique solution and find the solution. Can Theorem 5.4 be applied in each case?
a. y' = et−y, 0≤ t ≤ 1, y(0) = 1.
b. y' = t−2(sin 2t − 2ty), 1≤ t ≤ 2, y(1) = 2.
c. y' = −y + ty1/2, 2≤ t ≤ 3, y(2) = 2.
d. y' = (ty + y)/(ty + t) , 2≤ t ≤ 4, y(2) = 4.