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A tricky problem I've found amongst past papers for a class I'm taking.


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.


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.

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up vote 1 down vote accepted

Suppose that $x(t)$ is a solution periodic of period $T>0$. Multiply the equation by $x'(t)$ and integrate between $0$ and $T$. Since $$ \int_0^Tx''(t)x'(t)dt=\frac12\,(x'(t))^2\Bigr|_0^T=0\quad\text{and}\quad \int_0^T\sin(x(t))x'(t)dt=\cos(x(t))\Bigr|_0^T=0, $$ we get $$ \int_0^T\bigl((x'(t))^2+\epsilon\,(x'(t))^4 \bigr)dt=0, $$ so that $x'(t)\equiv0$ and $x(t)$ is constant. Thus, the only periodic solutions are the stationary solutions $x(t)=k\,\pi$.

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