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Letting $\vec{r}=\begin{pmatrix}x(\theta)\\y(\theta)\end{pmatrix}$ and that $\vec{v}^\perp$ denote a vector perpendicular to the vector $\vec{v}$. Then are there values of $c,d$, with $$\vec{r}''+c(\vec{r}'^\perp)+d{1\over |\vec{r}|^3}\vec{r}=0,$$ describing a circle for large enough $\theta$?

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    What's the source of the problem? And you do realize that your definition of $\vec{v}^\perp$ doesn't fix $\vec{v}^\perp$ uniquely?2012-04-01

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