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!