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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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    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.2012-01-06

1 Answers 1

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I see nothing wrong. A Picard iteration does not necessarily give you the first few terms of the Taylor expansion of the solution exactly. It only gives you approximate solutions.

If you do one more interationfor your problem, however, you'll see it contains the forth term.