The [tex]\dfrac{\mathrm dy}{\mathrm dx}[/tex] row is filled by just plugging in the corresponding values of [tex]x[/tex] and [tex]y[/tex] into the ODE.
For example, when [tex]x=-1[/tex] and [tex]y=1[/tex], you have
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{xy}3\implies\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=-1,y=1}=\dfrac{(-1)(1)}3=-\dfrac13[/tex]
and so on.
