Series solutions — a smooth but non-analytic forcing term

Consider the following statements:

(P) The initial value problem $y'+y=e^{-x^2}$, $y(0)=0$ has a Taylor series solution about the point $x=0$.

(Q) The initial value problem $y'+y=r(x)$, $y(0)=0$, where $r(x)=e^{-1/x^2}$ for $x\neq0$ and $r(0)=0$, has a Taylor series solution about the point $x=0$.

Then which of the following statements are true?