Use picards method to obtain the next four successive

Question

Under certain assumptions on f and Φ_o(x), the sequence {Φ_n(x) is known to converge to a solution

(a) Use Picard's method with Φ_o(x) ≡ 1 to obtain the next four successive approximations of the solution to

y' (x) = y(x) , y(0) = 1 .

Show that these approximations are just the partial sums of the Maclaurin series for the actual solution e^x.

(b) Use Picard's method with Φ₀(x) ≡ 0 to obtain the next three successive approximations of the solution to the nonlinear problem

 y'(x) = 3x - [y(x)]² , y(0) = 0 .

Graph these approximations for 0  x ≤ 1.

Initial value problem

y'(x) = 3[y(x)]^2/3 , y(2) = 0

does not have a unique solution. Show that Picard's method beginning with Φ₀(x) ≡ converges to the solution y(x) ≡ 0, whereas Picard's method beginning with Φ₀(x) = x - 2 converges to the second solution y(x) = (x - 2)³.