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