For $y'=x-y^2$, Compute: $\varphi_3(\frac{1}{2})=|y(\frac{1}{2})-y_3(\frac{1}{2})|$. I'd really love your help with the following Differential equations problem: Given $y'=x-y^2$, $y(0)=0$, I am ask to compute three Picard iterations and the value of the error: $\varphi_3(\frac{1}{2})=|y(\frac{1}{2})-y_3(\frac{1}{2})|$. I got that $$y_3(x)=\frac{x^2}{2}-\frac{x^5}{20}+\frac{x^8}{160}-\frac{x^{11}}{4400}.$$
I tried to find some pattern for a series, but I couldn't find one. How can I compute $\varphi_3$?
Thank you very much!
 A: The given equation is a Riccati differential equation; it cannot be solved in terms of (integrals of) elementary functions. Therefore you will not be able to compute $\phi_3\bigl({1\over2}\bigr)$ exactly.  To get at least an estimate you  have to go back to the proof of the existence/uniqueness theorem. Now check what the general estimates obtained there give numerically, when you apply them in the case at hand.
A: For the answer to your question, follow the advice of Christian Blatter. However, you might be interested in the fact that an exact solution to your problem can be written down in terms of Airy functions and their derivatives.
The solution to your problem reads
$$
y(x) = \frac{\sqrt{3} \operatorname{Ai}'(x) + \operatorname{Bi}'(x)}
{\sqrt{3} \operatorname{Ai}(x) + \operatorname{Bi}(x)} .
$$ 
It is an easy exercise to check that it fulfills the boundary condition $y(0)=0$ as $\sqrt{3} \operatorname{Ai}'(0) = -\operatorname{Bi}'(0)$. Furthermore, the Airy functions fulfill the differential equation ($z=$Ai, Bi)
$$z''-xz = 0;$$
thus with $\xi=\sqrt{3} \operatorname{Ai}(x) +\operatorname{Bi}(x)$, 
$$y' = \frac{\xi''}
{\xi} - \frac{\xi'^2}{\xi^2} = \frac{x\xi}{\xi} - \frac{\xi'^2}{\xi^2}=x - y^2.$$
