If X ∼ N(0, σ2), find the pdf of Y = |X|. If X ∼ N(0, $σ^2$
), find the pdf of Y = |X|. 
So far I have 
$F_Y(y) = P(\lvert x \rvert < y) = P(-y < x < y) = F_X(y) - F_X(-y)$ 
but I don't know where to go from there

In fact, this distribution has a name: Half-normal distribution.
$$
F_Y(y) = F_X(y) - F_X(-y)\\
=F_X(y) - (1-F_X(y)) = 2F_X(y) - 1
$$
then differentiate for the pdf
$$f_Y(y) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp  \left( -\frac{y^2}{2\sigma^2} \right), \quad y>0$$
