In Exercise, use the approach of Exercise 11 to generate the first three Picard iterates for u and then use y(x) = u(x = x0) to obtain the first three Picard iterates for y. Also verify that the first three terms of the third Picard iterate for y is the third degree Taylor polynomial about x0 of the solution to the initial value problem at x0.
y′ = x2 + y2, y(–1) = 0
Exercise 11
Show that if u(x) is a solution to the initial value problem u′ = f(x + x0, u), u(0) = y0, then y(x) = u(x = x0) is a solution to the initial value problem y′ = f(x, y), y(x0) = y0.
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