Consider the differential equation,
$$p^2 - (\frac{x}{y} - \frac{y}{x})p - 1 = 0, p = y'$$
This problem can be tackled by solving the quadratic in $p$. I found out that $p=\frac{x}{y}, \frac{-y}{x}.$
Therefore,
$\frac{dy}{dx} = \frac{-y}{x} \implies xy = c \tag{1}\label{eq1}$
Or, $\frac{dy}{dx} = \frac{x}{y} \implies x^2-y^2 = c \tag{2}\label{eq2}$
Combining, we can write, $(xy - c)(x^2-y^2 - c) = 0$, which is the General Solution.
But I don't understand why we used the same arbitrary constant $c$ in both equations — $\eqref{eq1}$ and $\eqref{eq2}$? Shouldn't it be two different arbitrary constants?
There are two separate one-parameter families of solutions: $xy = c$ and $x^2 - y^2 = c$. Combining them as $(xy-c)(x^2-y^2-c)$ is just obfuscation. And there is no reason for the constants $c$ to be the same: there is no connection between them.