A tricky problem I've found amongst past papers for a class I'm taking.
Show that the differential equation
has no periodic solution if $\epsilon$ is positive.
I think this is probably an application of Bendixson-Dulac for plane autonomous systems, but I can't establish how to manipulate the equation into a form to which we can apply B-D. Equally, it may call upon something different (although forcing a constant positivity for $\epsilon$ strongly suggests it isn't). Any assistance is appreciated. Regards as always, MM.