# Problem with Picard Iteration

I have

$\frac{dy}{dx} = y^2, y(0) = y_0$

I have solved this as

$y = \frac{y_0}{1 - x y_0}$

Which has the Taylor expansion

$y_0+y_0^2 x+y_0^3 x^2+y_0^4 x^3+y_0^5 x^4+ ...$

However, when I perform Picard iteration, I get:

$Iteration 0: y = y_0$

$Iteration 1: y = y_0 + \int y_0^2 = y_0 + y_0^2x$

$Iteration 2: y = y_0 + \int (y_0 + y_0^2x)^2 = y_0 + y_0^2x + y_0^3x^2 + \mathbb{\frac{y_0^4x^3}{3}}$

Everything is of the right order, there is just a factor of 1/3 at the end, where am I going wrong?

Thanks

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Do one more iteration... –  David Mitra Jan 6 '12 at 12:35
Ok so I did another iteration and I am still left with terms at the end off by a factor. However, the $x^3$ term is now correct. Can I then deduce that by taking limits all terms do this? Thanks –  T. Kiley Jan 6 '12 at 12:45
I don't think you're even guaranteed that $any\ iteration$ contains terms that are exactly the first few terms of the Taylor expansion of the solution. Of course, the iterates converge to the solution on some set. –  David Mitra Jan 6 '12 at 12:54