hi
Both of these differential equations are separable.
$\displaystyle \frac{dx}{dt} = -4xy$ may be rearranged as
$\displaystyle \frac{dx}{x} = -4y dt$,
and $\displaystyle \frac{dy}{dt} = 4xy-500y$ may be rearranged as
$\displaystyle \frac{dy}{y} = (4x - 500) dt$.
You may also compute $\displaystyle \frac{dy}{dx} = \frac{4xy - 500y}{-4xy}= -1 + \frac{125}{x}$ and then integrate that.