(a) Apply the iteration to y' = x + y, y(0) = 0. Also solve the problem exactly.

(b) Apply the iteration to y' = 2y², y(0) = 1. Also solve the problem exactly.

(c) Find all solutions of y' = 2√y, y(1) = 0. Which of them does Picard’s iteration approximate?

(d) Experiment with the conjecture that Picard’s iteration converges to the solution of the problem for any initial choice of y in the integrand in (7) (leaving y₀ outside the integral as it is). Begin with a simple ODE and see what happens. When you are reasonably sure, take a slightly more complicated ODE and give it a try.