Explicit solutions to a one parameter family of solutions The first given equation is
$$8x^2-y^2=C$$
and differentiating this verifies that it is equal to $\frac{ydy}{dx}=8x$ which is our differential equation
$$16x-2yy'=0$$
$$yy'=8x$$
I'm having trouble finding explicit solutions in each of the four quadrants, how do I continue to find the solutions and their respective intervals?
 A: Integrating the differential equation recovers your original form.
The family of curves (as $C$) varies give a family of hyperbolae. For example $C=-1$ gives

and $C=2$ gives

Alll the curves are symmetrical about both axes, so $$y=\sqrt{8x^2-C}\text{ with }C=-1$$ gives the solution to the differential equation in the upper half-plane (ie 1st and 2nd quadrants). Note that there are no values for small $|x|$. Similarly, $$y=-\sqrt{8x^2-C}$$ gives the solution in the lower half-plane. 
Similarly for the second graph with $C=2$. This time both solutions ($+\sqrt{}$ and $-\sqrt{}$) work for all $x$, but there is a gap in the $y$ values.

Added later.
As usual with solution curves for ODE the curves for the family of solutions fill the plane. The plot below with curves for a range of values of $C$ from -20 to +50 may make that a little clearer.

Note the degenerate conic, which is a pair of intersecting lines, that are also the asymptotes for the other solutions.
A: from $yy'=8x$ we have
$$
ydy=8xdx
$$
we can write the solution $y^2=8x^2+C$, that is an implicit function, or in the form
$$
y=\pm \sqrt{8x^2+C}
$$
that is a couple of explicit functions.
