I am trying to graph the direction field and isoclines for $\dfrac{dy}{dx} = \dfrac{-y}{x}$. From the direction field, it can be seen that there are numerous (partial) isoclines that pass through the origin. However, the origin is $\dfrac{dy}{dx} = \dfrac{0}{0}$ -- an indeterminate form. This is why I described the isoclines that pass through the origin as "partial" -- they have constant slopes and take the form of linear curves through the origin, but, at the origin, the isocline is broken, since we have $\dfrac{dy}{dx} = \dfrac{0}{0}$.
So how do I graph these isoclines through the origin if the origin is $\dfrac{dy}{dx} = \dfrac{0}{0}$? Wouldn't this point mean that they aren't actually isoclines, since isoclines are curves connecting points of constant slope $\dfrac{dy}{dx} = m$?
I would greatly appreciate it if people could please take the time to clarify this.
You don't graph them through the origin. Rather, you graph them up to (but not including) the origin, where they necessarily terminate, as the function is undefined there. One could think of this as having numerous isoclines radiating away from a common source.
Remember, it's okay for isoclines not to be connected. Consider instead $\frac{dy}{dx}=x^2,$ and consider any positive slope.