I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function $\varphi (x,y)$ such that the expression $$ \frac{\partial (f \varphi)}{\partial x} + \frac{\partial (g \varphi)}{\partial y} $$ has the same sign for all $x, \; y$ in the domain.
My question is, how is the function $\varphi (x, \; y)$ (Dulac's Multiplier) found?
I the solutions to the questions given out in lectures, this usually takes values $$ \frac{1}{xy} \quad \text{or} \quad x^{n} $$ where $n$ is some parameter to be found. Can $\varphi$ take any other values?
Yes, according to the theorem, $\varphi(x,y)$ can be any continuously differentiable function. There is no general method to find suitable $\varphi(x,y)$, given two functions $f$ and $g$. Usually, you'll get an idea about a $\varphi(x,y)$ that might work when looking at the PDE $\frac{\partial (f \varphi)}{\partial x} + \frac{\partial (g \varphi)}{\partial y}$. Note that you can only hope to find such a function if the planar system has no periodic orbits; therefore, if you fail to find a suitable $\varphi(x,y)$, that could also mean that the system at hand does possess a periodic orbit somewhere.
So, to conclude, if you're able to find such a $\varphi(x,y)$ for a given planar system, that's good, but it's often a matter of chance. A bit similar to the Lyapunov function.