A tricky problem I've found amongst past papers for a class I'm taking.
Problem
Show that the differential equation
$\frac{d^2x}{dt^2}+\frac{dx}{dt}+\epsilon \bigg(\frac{dx}{dt}\bigg)^3+\sin(x)=0$
has no periodic solution if $\epsilon$ is positive.
Thoughts
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.