Consider the initial value problem
$\cases{y'(x)=xy(x)+2,\\ y(0)=3}$.
Give a recursive formula for the coefficients of the Maclaurin series of a solution $\varphi$ of the above initial value problem.
I've tried to establish some recursive rule by which the derivaties of $\varphi$ form and obtained that $\varphi'(0)=2,\varphi''(0)=3,\varphi'''(0)=4,\varphi^{(4)}(0)=9,\varphi^{(5)}(0)=16$ and $\varphi^{(6)}(0)=45$ which doesn't look as if they form recursively, even if we consider $\frac{\varphi^{(i)}(0)}{i!}$.