Question

Use the differential equation given by dy/dx=xy/3, y>0 complete the table of values

Answer 1

The $\dfrac{\mathrm dy}{\mathrm dx}$ row is filled by just plugging in the corresponding values of $x$ and $y$ into the ODE.

For example, when $x=-1$ and $y=1$, you have

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{xy}3\implies\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=-1,y=1}=\dfrac{(-1)(1)}3=-\dfrac13$

and so on.