Variations of Dawson's function

I am studying Dawson's function : $\displaystyle F : x \mapsto e^{-x^2}\int_0^x e^{t^2} dt$.

I would like to prove that $F$ attains a maximum at a certain value $x_0 \in (0,1)$, and is increasing over $[0,x_0]$ and decreasing over $[x_0, +\infty)$.

The only thing I managed to prove about this function is that $\displaystyle\lim_{x \rightarrow +\infty} F(x)=0$, which gives the existence of a maximum over $[0,+\infty)$, but does not help to determine the variations of $F$.

I also know that $F$ is solution of the differential equation $y'+2xy=1$, but I cannot see how it could help.

So the question is : how to determine the variations of $F$ ? (and bonus : how to prove that $x_0 <1$ ?)